ESAIM: Probability and Statistics
January 2003, Vol. 7, 23–88
DOI: 10.1051/ps:2003008
SUPERPOSITION OF DIFFUSIONS WITH LINEAR GENERATOR
AND ITS MULTIFRACTAL LIMIT PROCESS ∗
Endre Iglói 1 and György Terdik 1
Abstract. In this paper a new multifractal stochastic process called Limit of the Integrated
Superposition of Diffusion processes with Linear differencial Generator (LISDLG) is presented which
realistically characterizes the network traffic multifractality. Several properties of the LISDLG
model are presented including long range dependence, cumulants, logarithm of the characteristic
function, dilative stability, spectrum and bispectrum. The model captures higher-order statistics
by the cumulants. The relevance and validation of the proposed model are demonstrated by real
data of Internet traffic.
Mathematics Subject Classification. 62M10, 60J60, 60G10, 93E12.
Received May 14, 2002.
Contents
Introduction
1. Multifractality, self-similarity and dilative stability
2. Diffusion processes with a linear differential generator (DLG processes)
2.1. Characterization of the DLG process
2.2. Cumulants, characteristic functional and spectra of the DLG process
3. Superposition of DLG processes and its multifractal limit process
3.1. Superposition of random-parameter stationary DLG processes (SDLG process)
3.2. Multifractal limit processes (LISDLG and ∆LISDLG processes)
3.3. Spectra of the SDLG process and its offsprings
3.4. Nonlinearity of the ∆ISDLG and ∆LISDLG processes
3.5. Relation with [1], a similar result of Barndorff–Nielsen
4. Application to computer network traffic data
4.1. The dilative stability and parameter estimation
4.2. Fitting of distributions and spectra
5. Open problems and directions of future research
Appendix A. Multivariate Γ distribution
References
24
25
27
27
33
39
39
46
56
72
73
76
76
79
81
84
87
Keywords and phrases: Fractals, long range dependence, self-similarity, stationarity, higher order statistics, bispectrum,
network traffic, superposition, diffusion processes, CIR process, DLG process, square root process.
∗
This research is supported by the Hungarian National Science Foundation OTKA No. T 032658.
1
Institute of Mathematics and Informatics, University of Debrecen, 4010 Debrecen, PF 12, Hungary;
e-mail: [email protected]
c EDP Sciences, SMAI 2003
24
E. IGLÓI AND C. TERDIK
Introduction
Computer network traffic has recently been the subject of various types of statistical studies including
fractal analysis, and in particular, measuring and modeling Long-Range Dependence (LRD), investigating
self-similarity [26] and showing multifractal properties [39]. The common agreement among several empirical
findings about the general properties of traces is summarized in [31] as follows:
• many signals show significant LRD, but behavior inconsistent with strict self-similarity;
• for many signals, the scaling behavior of moments as the signal is aggregated is a nontrivial function
of the moment order;
• many signals have increments that are inherently positive, skewed and hence non-Gaussian.
There are some additional properties motivated by our experimental study of ATM traces, see [41], providing
strong evidence of the presence of Γ distribution and real-valued bispectrum. Therefore there are two
additional requirements:
• the marginal distribution of signals of ATM traces is close to Γ distribution;
• signals of ATM traces have a real-valued bispectrum.
Having these properties in mind we will study in the present paper a certain nonlinear diffusion process,
superposition of such processes with random coefficients, the limit of the centralized integral processes of the
superposition processes and its increment process. We will consider a possible application too. The main
objective is to find a multifractal model which has an analytically and statistically tractable higher order
cumulant structure.
The paper starts with the introduction. In Section 1 we introduce the notions of multifractality, selfsimilarity and dilative stability. In Section 2 the basic process of superposition is considered. We call it
Diffusion with Linear differential Generator (DLG). We summarize the results obtained by Watanabe and
his coauthors concerning the connection between the DLG process and infinitely decomposable diffusion
processes and continuous-state branching processes. It is also pointed out that the finite-dimensional distribution of the DLG process is multivariate Γ. We are concerned with stationary observations, therefore the
stationary DLG process will be studied in greater detail. The exact cumulants, characteristic functional and
spectra of the stationary DLG process will be given.
In Section 3 we will consider a triangular array of random coefficient DLG processes. It will be shown that
the row sums converge to a limit process called Superposition of DLG processes (SDLG process). As for the
properties of the SDLG process we will prove that the logarithm of the characteristic functional of the SDLG
process is the mixture of the logarithm of the characteristic functionals of the component DLG processes.
The SDLG process also has a Γ finite-dimensional distribution. It is stationary, it has a real-valued positive
bispectrum and it is LRD. The main result of the paper is the existence of the Limit of Integrated SDLG
processes, referred to as the LISDLG process. It is shown that both the LISDLG process and its increment
process, labeled as ∆LISDLG, are multifractals. The covariance structure of the LISDLG process is the same
as that of the Fractional Brownian Motion (FBM). Consequently, the spectrum of the ∆LISDLG process is
the same as that of the discrete time Fractional Gaussian Noise (FGN), ∆B (H) (k) = B (H) (k + 1) − B (H) (k),
k ∈ Z. The bispectrum of the ∆LISDLG process is also given, and it has the extraordinary property that
it is real-valued. If the bispectrum of a linearly regular long-range dependent discrete time process is real
(non-zero), then the process must be nonlinear. Hence, the ∆LISDLG process is nonlinear. At the end of
Section 3 we compare our results with those of [1]. In this context it is pointed out that the Ornstein–
Uhlenbeck (OU) type models considered in [1] cannot fulfil our requirements for the bispectrum to be real.
The logarithm of the m-th order cumulants of the LISDLG process scales linearly with the logarithm of
time. A similar scaling behavior also holds for the ∆LISDLG process. In particular (1.9) below means that
the logarithm of the m-th order cumulants of the series averaged at level n depends linearly on log n with
coefficient 2(H − 1) being independent of m, where H is the Hurst parameter. This fact is more special than
the multifractal property, nevertheless it holds under more general circumstances than our case, e.g. for the
OU type models of [1], see Section 3.5. Moreover, the LISDLG process has an interesting property, which
the authors call dilative stability, which implies the above mentioned scaling behavior (1.9).
In Section 4 we will apply our ∆LISDLG model to real data. The time series of ATM traces measured in
SUNET fits our model very well. The feasibility of carrying out parameter estimation utilizing the dilative
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
25
stability is also discussed to some extent. In Section 5 some open problems and possible research directions
are suggested.
An Appendix giving some basic facts of multivariable Γ distribution closes the paper.
1. Multifractality, self-similarity and dilative stability
A wide variety of physical systems including data network traffic exhibit fractal properties. We are
interested in fractal data, i.e. data “look the same across a wide range of scales”, at least in certain respect.
The notion of fractal, in the sense that it is similar on all scales, is translated into the stochastic analysis by
the definition of Self-Similar process with Stationary Increments, see [23]. We will denote it by sssi. The
stochastic process Y (t) is called H-sssi, if it has stationary increments and, for all a > 0 real numbers,
d
Y (at) = aH Y (t) ,
d
where = means the equality of finite-dimensional distributions. The parameter H is referred to as the Hurst
parameter or Hurst exponent. An example is the FBM B (H) (t), where parameter H ∈ (0, 1), see [2, 29, 33].
Let m ∈ N and suppose that the m-th order cumulant, cumm (Y (t)), exists. Now it follows from selfsimilarity that
cumm (Y (t)) = tmH cumm (Y (1)) ,
(1.1)
i.e. the logarithm of cumm (Y (t)) scales linearly with log(t) and the coefficient is the homogeneous linear
function (of m) mH. Note that [39] considers absolute moments instead of cumulants, i.e.
m
E |Y (t)|
m
= tmH E |Y (1)|
(1.2)
instead of (1.1). For H-sssi processes both (1.1) and (1.2) hold.
There are different generalizations of the self-similar processes to multifractal processes. One concerns the
almost sure fractal property of continuous paths according to the deterministic theory of fractals, see [18]. The
other one is based on the statistical inference of the process. We follow this latter direction. A generalization
of the scaling (1.2) is
log E |Y (t)|
m
= mH (m) log (t) + c (m) ,
(1.3)
where the dependence of H on m points on the multifractal behaviour of the process. The papers [4] and [39]
define the notion of multifractal essentially by (1.3). We prefer cumulants rather than absolute moments for
describing the multifractal property because the scaling properties should not change with additive constants
and with summing up of independent copies of a process.
Definition 1. A stochastic process Y (t) is called multifractal if, for each m ∈ N the cumulant cumm (Y (t))
exists and the logarithm of |cumm (Y (t))| scales linearly with log (t), and the coefficient is mH (m), i.e.
log (|cumm (Y (t))|) = mH (m) log (t) + c (m) .
(1.4)
Note that in the above definition constant c (m) is possibly −∞, if the corresponding cumulant is zero; in
this case (1.4) holds with arbitrary H (m). Note also that H (m) = H for H-sssi processes.
The H-sssi property of Y (t) implies that the increment process
X(k) = Y (k + 1) − Y (k) ,
k ∈ Z,
(1.5)
which is stationary, is characterized in terms of the behavior of the aggregated processes arising by averaging
X(k) over nonoverlapping blocks of size n ∈ N, i.e.
X (n) (k) =
n−1
1X
X(kn − j),
n j=0
k ∈ Z,
(1.6)
26
E. IGLÓI AND C. TERDIK
have similar finite-dimensional distributions, specifically,
d
∀n ∈ N : X (n) (k) = nH−1 X(k).
(1.7)
The stationary process X(k) fulfilling (1.7) is called Stationary Self-Similar with Hurst exponent H, i.e. an
H-sss process. A typical example is the FGN process X(k) $ ∆B (H) (k) = B (H) (k + 1) − B (H) (k), k ∈ Z,
which is a unique Gaussian H-sss process, see [22, 37]. The m-th order cumulants of the aggregated H-sss
processes also scale as
(1.8)
cumm X (n) (k) = nm(H−1) cumm (X(k)) .
Now if a stationary process X(k) fulfils equation (1.8) for each n ∈ N and m ∈ N, then the logarithm
of the modulus of cumm X (n) (k) scales linearly with log (n) with coefficients m (H − 1) depending on m
homogeneous linearly. Therefore such processes will be called unifractal processes because the Hurst exponent
H is universal for all n ∈ N and m ∈ N. The assumption that a process Y (t) is H-sssi is of course stronger
than the assumption that the increment process is unifractal. One can go even further in weakening the
assumption.
Definition 2. A stationary process X(k), k ∈ Z, is multifractal if
log cumm X (n) (k) = m (H(m) − 1) log (n) + c (m) ,
for all m ∈ N, i.e. allowing the exponent H to vary with the order m.
Similarly to Definition 1, constants c (m) may be −∞.
1
A particular class of multifractal processes is the set of p 2(H−1) -dilative stable processes which have
cumulants of all orders.
Definition 3. We say that a process Y (t) is dilative stable with scaling function c : R+ → R+ (or cdilative stable, for short), if for every p > 0, k ∈ N, and t1 , . . . , tk > 0, the characteristic function ϕ of the
k-dimensional distribution satisfies
p
1
ϕ(Y (t1 ),... ,Y (tk )) (u1 , . . . , uk ) = ϕ c(p)
(Y (c(p)t1 ),... ,Y (c(p)tk )) (u1 , . . . , uk ),
for all u1 , . . . , uk ∈ R.
Note that the attribute “dilative” alludes to the dilatation (t1 , . . . , tk ) 7→ c(p) (t1 , . . . , tk ) in the time
1
domain. If cumulants of all orders of Y (t) exist, then the c(p) = p 2(H−1) -dilative stability implies that
m
1
p cumm (Y (t)) = p 2(1−H) cumm Y p 2(H−1) t ,
and taking p := t2(H−1) and t := 1, we have
log (|cumm (Y (t))|) = (2 (H − 1) + m) log t + log (|cumm (Y (1))|) ,
i.e.
Y (t) log cumm
= 2 (H − 1) log t + c (m) .
t
Similarly, it holds for the aggregated process (1.6) of the increment process (1.5), that
log cumm X (n) (k) = 2 (H − 1) log (n) + c (m) .
(1.9)
What is interesting is that the coefficients of log t and log n do not depend on m. However, this must not be
confused with the unifractal property; it is an outcome of the dilative stability.
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
27
The increments of an H-sssi process Y (t) have another particular second-order property called LongRange Dependence (LRD). The definition of LRD for a stationary process X(k), k ∈ Z, is that the autocovariance series r(k) $ cov (X(i), X(i + k)) = cum2 (X(i), X(i + k)), k ∈ Z, behaves asymptotically
as
r(k) ' L(k)k 2(H−1) ,
k→∞
(1.10)
where L(x) is a slowly varying function at infinity, and 1/2 < H < 1. Note that the slowly varying function
has not a significant role. A stationary process is asymptotically second-order self-similar, see [8], if the
relationship
(1.11)
∀n ∈ N : r(n) (k) = cum X (n) (i), X (n) (i + k) = n2(H−1) r(k)
holds asymptotically as n → ∞ and k → ∞, or more precisely, the series of the autocorrelation functions
ρ(n) (k) = r(n) (k)/r(n) (0) converges, as n → ∞, to an autocorrelation series ρ(∞) (k), k ∈ Z, and ρ(∞) and
ρ(1) are equivalent at infinity, i.e. ρ(∞) (k) and ρ(1) (k) converge to zero in the same order as k → ∞, i.e.
ρ(∞) (k) ' const.ρ(1) (k).
k→∞
This type of self-similarity is the weakest and it is the one which is implied by LRD. For an example see the
ΓMOU process in [16]. The main subject of this paper, the ∆LISDLG process is not self-similar, but as it
is LRD, it is asymptotically second-order self-similar.
2. Diffusion processes with a linear differential generator
(DLG processes)
2.1. Characterization of the DLG process
Let us consider the diffusion-type stochastic differential equation
p
dR(t) = (µ + 2αR(t)) dt + 2σ R(t)+ dB(t)
(2.1)
.
where R(t)+ = max (R(t), 0), moreover µ > 0, α ∈ R, σ > 0 are the parameters, and B(t) is a Brownian
motion. The equation (2.1) has a unique strong solution R(t), see the paper [43] or Theorem IV.3.2 in [17]
about the existence and uniqueness of the solution. It is not hard to prove that solution R(t) is conservative,
because P (R(t) > 0 | R(0) > 0) = 1 for all t > 0, see Example IV.8.2 in [17]. Assuming that R(0) > 0 one
could omit the notation of the nonnegative part in (2.1), replacing R(t)+ by R(t), i.e.
dR(t) = (µ + 2αR(t)) dt + 2σ
p
R(t)dB(t).
(2.2)
Put
δ$
µ
·
σ2
(2.3)
The solution R(t) of the stochastic differential equation (2.2) will be treated below under the name δdimensional Diffusion process with a Linear differential Generator (DLG process), because its differential
generator A is given by
A = (µ + 2αx)
∂2
∂
+ 2σ 2 x 2 ,
∂x
∂x
and both coefficients of A are linear, see [11]. Note here that the DLG process has recently been used
successfully for modeling interest rates, because it is one of the simplest models avoiding negative values.
There is a lack of agreement on terminology: after the authors of [9], for example, reference [36] would
28
E. IGLÓI AND C. TERDIK
label this as CIR process and in the special case α = 0 (we will have mostly α < 0) it would be called the
squared Bessel process. Because of the form (2.1) it is called square-root diffusion, too. In the mathematical
literature this process has not been denominated although it was mentioned by Kolmogorov in 1931, see [21]
and was studied intensively by [12, 19] and by Watanabe in the seventies, see below.
Albeit at first sight the DLG process seems to be an overly special process, in fact the set of DLG processes
is a parametric class of substantially general processes.
Proposition 1. If α < 0, and δ ∈ N, then the DLG process is just the squared radial part of a particular
δ-dimensional OU process.
Proof (by Chap. 31 of [36]). Let us take the OU processes Xi (t), i = 1, . . . , δ, solutions of the Langevin
equations
dXi (t) = αXi (t)dt + σdBi (t),
(2.4)
where Bi (t), i = 1, . . . , δ, are independent Brownian motions. Let
R(t) $
δ
X
Xi (t)2 .
i=1
Then Itô’s formula implies
δ
X
p
Xi (t)
p
dBi (t).
dR(t) = 2αR(t) + δσ 2 dt + 2σ R(t)
R(t)
i=1
Since
B(t) $
δ Z
X
i=1 0
t
Xi (s)
p
dBi (s)
R(s)
is a continuous square-integrable local martingale (with respect to the product σ-field) with quadratic variation t, it follows by Lévy’s theorem (see, e.g. Th. II.6.1 in [17]) that B(t) is a Brownian motion. Thus, R(t)
fulfils (2.2) with µ = δσ 2 .
Let α ∈ R and σ > 0 be arbitrarily fixed parameters.
Notation 1.
Rx(δ) (t) $ solution of (2.2) with µ = δσ 2 starting from the initial value x,
(δ)
i.e. Rx (0) = x, and
Px(δ) $ distribution of Rx(δ) on C[0, ∞)
for x > 0 and δ > 0. Sometimes, when it is not important to feature the superscript (δ), we will omit it.
Proposition 2.
(δ +δ )
Px(δ11 ) ~ Px(δ22 ) = Px11+x22 ,
for arbitrary δ1 , δ2 > 0 and x1 , x2 > 0, where ~ denotes the convolution.
Proof. Let
q
(δ )
dRx(δii ) (t) = δi σ 2 + 2αRx(δii ) (t) dt + 2σ Rxii (t)dBi (t),
(2.5)
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
29
i = 1, 2, be the defining equations of the two DLG processes, where B1 (t) and B2 (t) are independent
Brownian motions. Summing the two equations up, we have
q
(δ )
(δ )
(δ1 )
(δ2 )
2
(δ1 )
(δ2 )
d Rx1 (t) + Rx2 (t) = (δ1 + δ2 ) σ + 2α Rx1 (t) + Rx2 (t) dt + 2σ Rx11 (t) + Rx22 (t)
q
q


(δ )
(δ )
Rx11 (t)
Rx22 (t)
dB1 (t) + q
dB2 (t) .
(2.6)
× q
(δ1 )
(δ2 )
(δ1 )
(δ2 )
Rx1 (t) + Rx2 (t)
Rx1 (t) + Rx2 (t)
Define
.
B(t) =
Zt
0
q
(δ )
Rx11 (s)
q
(δ )
Rx22 (s)
Zt
q
dB1 (s) +
(δ )
(δ )
Rx11 (s) + Rx22 (s)
0
q
dB2 (s).
(δ )
(δ )
Rx11 (s) + Rx22 (s)
(2.7)
Then one can use the same argument as at the end of the proof of Proposition 1 to get that B(t) is a
Brownian motion. Finally, let
.
(δ +δ )
Rx11+x22 (t) = Rx(δ11 ) (t) + Rx(δ22 ) (t).
(δ +δ )
Then we have from (2.6) and (2.7) that Rx11+x22 (t) is really a DLG process with dimension δ1 + δ2 and
(δ )
(δ )
starting from x1 + x2 . Since processes Rx11 (t) and Rx22 (t) are independent, we obtain (2.5).
Notation 2. Let P(δ) be the equivalence class of distributions of δ-dimensional DLG processes, i.e.
n
o
P(δ) $ Px(δ) : x > 0 ,
(δ)
for δ > 0. It is natural to call P(δ) the distribution class of Rx . Also define the parametric set
o
n
P $ P(δ) : δ > 0 ·
o
n
(δ)
Obviously, P is a classification of the set Px : δ > 0, x > 0 and it is compatible with respect to the
convolution operation, so one can naturally extend the convolution to P via class representatives, i.e. with
the aid of (2.5), as follows:
Definition 4. Let δ1 , δ2 > 0. Then
P(δ1 ) ~ P(δ2 ) $ P(δ1 +δ2 ) .
(2.8)
Writing formula (2.5) in the language of Laplace transforms of finite dimensional distributions, and taking
particular initial values, we have
(δ )
(δ )
(δ +δ )
Lcδ11 (u)Lcδ22 (u) = Lc(δ11 +δ22 ) (u),
(2.9)
(δ)
the Laplace transform at point u ∈ Rk+ of the k-dimensional
for every c, δ1 , δ2 > 0, where Lx (u) denotes
(δ)
(δ)
distribution of Rx (t1 ), . . . , Rx (tk ) , and k ∈ N and t1 , . . . , tk ∈ R+ are arbitrary but fixed. Because
Cauchy’s multiplicative functional equation has a unique solution in the set of measurable functions, the
solution of (2.9) is
δ
(δ)
,
Lcδ (u) = L(1)
c (u)
(2.10)
30
E. IGLÓI AND C. TERDIK
δ > 0. Considering (2.10) for fixed c and δ and for every k ∈ N and every u ∈ Rk+ , we have
~δ
(δ)
,
Pcδ = Pc(1)
(2.11)
where the convolution power is defined as the inverse Laplace transform of the right hand side of (2.10).
This notion of convolution power is, clearly a generalization of the one with positive integer exponent. One
could naturally extend it to P via class representatives, however, the following definition is sufficient for our
purposes.
Definition 5. Let c, δ > 0.
P(1)
~δ
.
= the distribution class of
Pc(1)
~δ
.
Obviously, this definition is right, because by (2.11), the class is independent of c. Now we formulate the
statement (2.11) in a concise form as follows:
Proposition 3.
~δ
P(δ) = P(1)
(2.12)
holds for every δ ∈ R+ .
As mentioned above, the set of DLG processes is a parametric class of substantially general processes. To
explain it we introduce some notions.
Definition 6 (see [34]). A stochastic process is infinitely decomposable, if for every n ∈ N its distribution
class P (i.e. the set of its distributions for all possible initial values) can be represented in the convolution
power form
~n
P = (Pn )
,
with some proper distribution classes Pn .
Remark 1. Clearly (2.8) implies that the DLG process is infinitely decomposable.
Definition 7. A process X(t) is a Continuous-state Branching process with Immigration (CBI process)
with a.s. continuous trajectories, if X(t) is a diffusion process with nonnegative trajectories, and there exist
functions f (t, u) > 0 and g (t, u) > 0, t, u > 0, such that the conditional Laplace transform of X(t) has the
form
(2.13)
E e−uX(t) | X(0) = x = e−f (t,u)x g (t, u) ,
for all x > 0.
We now list without proofs some statements from Section 1 of [34], which indicate the ponderosity of the
DLG process.
Theorem 1 (see [34]). The following statements are equivalent:
1) R(t) is a DLG process;
2) R(t) is an infinitely decomposable diffusion process with nonnegative trajectories;
3) R(t) is a CBI process with a.s. continuous trajectories.
Theorem 1 is the elegant, and the following Theorem 2 is the exact version.
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
31
Theorem 2 (see [34]). The following statements are equivalent:
1)
(δ)
P : δ > 0 is a parametric set of the distribution classes of DLG processes with the same parameters
α ∈ R and σ > 0;
2)
(δ)
P : δ > 0 is a parametric set of the distribution classes of diffusion processes with nonnegative
trajectories, and it is a one-parameter convolution semigroup, i.e. (2.8) holds for all δ1 , δ1 > 0;
3)
(δ)
P : δ > 0 is a parametric set of the distribution classes of a.s. continuous CBI processes and
(2.12) holds for all δ > 0.
Observe that (2.12) also follows from Theorem 2.
We now examine the finite-dimensional distributions of the DLG process. Using the following result of
Section 1 in [34], one can directly calculate the conditional characteristic function.
Lemma 1 (see [34]). Let X(t) be a CBI process with a.s. continuous trajectories, and let the differential
generator, corresponding to it by Theorem 1, be of the form
(µ + 2αx)
∂2
∂
+ 2σ 2 x 2 ·
∂x
∂x
Then functions f (t, u) and g (t, u) in (2.13) are such that f (t, u) is the solution of the differential equation
∂
f (t, u) = −2σ 2 f 2 (t, u) + 2αf (t, u)
∂t
f (0, u) = u,
(2.14)
and

g (t, u) = exp −µ
Zt

f (s, u) ds .
(2.15)
0
Proposition 4. The conditional characteristic function of the instantaneous value of the DLG process,
given its initial value, is
(δ)
ϕR(δ) (t) u | R(δ) (0) = x $ E eiuR (t) | R(δ) (0) = x
!
δ
−2
σ2
iuxe2αt
2αt
1−e
1 − iu
,
(2.16)
= exp
σ2
−α
1 − iu −α
(1 − e2αt )
u ∈ R.
Proof. By Theorem 2 it is enough to compute the conditional characteristic function of a CBI process with
e(δ) (t) be such a process. Then the conditional
a.s. continuous trajectories and with property (2.12). Let R
e (δ) (t) is of the form
Laplace transform of R
e(δ) (0) = x
e(δ) (0) = x $ E e−uRe(δ) (t) | R
ψRe(δ) (t) u | R
= e−f (t,u)x g(t, u),
(2.17)
and by Lemma 1 f (t, u) and g (t, u) are given by (2.14) and (2.15). The solution of (2.14) can be easily found
by the replacement h = 1/f to be
f (t, u) =
1−
ue2αt
·
(1 − e2αt )
2
u σα
32
E. IGLÓI AND C. TERDIK
Hence, by (2.15),
g(t, u) =
δ
−2
σ2
1 − e2αt
.
1−u
α
Thus, by (2.17) the Laplace transform
e(δ) (0) = x = exp
ψRe(δ) (t) u | R
−uxe2αt
σ2
1 + u −α
(1 − e2αt )
!
δ
−2
σ2
2αt
1+u
,
1−e
−α
whereby we get the characteristic function in the statement.
Formula (2.16) can be obtained also in a more direct probabilistic way, see [17], IV, 8, formula (8.10) and
below.
The transition density of the DLG process can be derived following [19], see also [12]. We quote only the
resulting formula:
.
Proposition 5. If α < 0, then the transition density px (t, y) = pR(δ) (t)|R(δ) (0) (y|x) of the DLG process is
−α
1
exp
px (t, y) = 2 ·
σ
1 − e2αt
α y + xe2αt
·
σ 2 1 − e2αt
x 2αt
e
y
−β/2
Jβ
!
p
xye2αt
2 |α|
·
,
σ2
1 − e2αt
(2.18)
.
where β = µ/ 2σ 2 − 1 and Jβ denotes the modified Bessel function of the first kind of order β.
To guarantee the stationarity of the processes under consideration, from now on we will restrict the set
of values of the parameter α to negative values, i.e. the condition α < 0 will hold in the rest of the paper.
Taking the limit as t → ∞ in (2.16), because α < 0, we have the characteristic function of the stationary
distribution
ϕR(δ) (t) (u) $ E e
iuR(δ) (t)
= lim ϕR(δ) (t) u | R
(δ)
t→∞
− δ2
σ2
(0) = x = 1 − iu
,
−α
i.e. the characteristic function of the Γ δ/2, −σ 2 /α distribution. The same stationary distribution could
be obtained by taking the limit as t → ∞ in (2.18). Thus we can state the following:
Proposition 6. The one-dimensional stationary distribution of the δ-dimensional DLG process R(t) defined
by (2.2), where α < 0, is Γ, i.e.
R(t) ∼ Γ
δ σ2
,
2 −α
·
In the rest of the paper we will deal with stationary DLG processes, i.e. we will assume that the stationary
distribution is attained. In other words, we will suppose that the process starts from its stationary onedimensional distribution.
Let us examine the m-dimensional distribution of the stationary DLG process.
Proposition 7. Let t $ (t1 , . . . , tm ), and t1 < . . . < tm . Then the m-dimensional stationary distribution
of the δ-dimensional DLG process R(t) defined by (2.2), where α < 0, is m-dimensional Γ, i.e.
(R
(δ)
(t1 ), . . . , R
(δ)
(tm )) ∼ Γ
δ
, 2Σ ,
2
33
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
where Σ is the m × m variance-covariance matrix of the accompanying Gaussian random vector (X(t1 ), . . . ,
X(tm )) and X(t) is the stationary OU process defined by
Zt
eα(t−s) dB(s).
X(t) = σ
(2.19)
−∞
(See the Appendix for the multi-dimensional Γ distribution.)
Proof. Because
distributions of
power rule connects the m-dimensional
of (2.12), the same convolutional
(δ)
(δ)
(1)
(1)
the vectors Rδx (t1 + s), . . . , Rδx (tm + s) and Rx (t1 + s), . . . , Rx (tm + s) . Letting s → ∞, we get
the power rule for the m-dimensional stationary distributions, i.e. denoting the stationary distribution of
(δ)
(R(δ) (t1 ), . . . , R(δ) (tm )) by Pt we have
(δ)
Pt
~δ
(1)
= Pt
.
(2.20)
(1)
is the distribution of a vector the components of which are the squares of the components of
(δ)
a homoscedastic Gauss–Markov vector, equation (2.20) implies that Pt is the m-dimensional Γ δ2 , 2Σ
distribution.
Remark 2. The fact that the above vector (R(δ) (t1 ), . . . , R(δ) (tm )) has the multi-dimensional Γ δ2 , 2Σ
distribution means a new proof for Theorem 1 in [14] about the infinitely divisibility of the Γ 12 , 2Σ distribution. See also the Appendix.
Since Pt
The following theorem is the stationary counterpart of Proposition 3, and it is a direct consequence
of (2.20). It is probably the most useful trait of the stationary DLG process, because it makes possible to
derive properties of the δ-dimensional DLG process from those of the simpler one-dimensional process.
Theorem 3. The distributions on C[0, T ] of the δ-dimensional and of the 1-dimensional stationary DLG
processes, i.e. P (δ) and P (1) are connected by the power rule
~δ
,
P (δ) = P (1)
δ > 0. Here ~δ denotes the convolution power, defined via the same convolution power of the finite dimensional distributions.
Remark 3. Clearly, Theorem 3 implies that the stationary DLG process is infinitely divisible.
Remark 4. A useful observation found in [7] is that for α < 0 the right-hand side of (2.16) is just the
characteristic function of the mixture of Γ distributions by Poissonian weights, namely,
Rx(δ) (t)
∼Γ
σ 2 (1 − e2αt )
δ
+ η,
2
−α
, where η ∼ Poi
−αx
σ 2 (e−2αt − 1)
·
(2.21)
The proof is a simple and short calculation, so we omit it. Equation (2.21) provides the simulation of the
DLG process once a discrete set of times has been chosen.
2.2. Cumulants, characteristic functional and spectra of the DLG process
Let R(t) be the stationary DLG process. We will consider R(t) on an interval [0, T ] where T > 0 is
arbitrary but fixed.
We are now interested in the cumulants of finite-dimensional projections, and characteristic functional of
the measure induced on C[0, T ] by the stationary DLG process R(t). Let us make it clear what we mean
by the characteristic functional of a process Z(t) with trajectories a.s. in C[0, T ]. We will also call it the
characteristic functional of the measure generated on C[0, T ] by process Z(t).
34
E. IGLÓI AND C. TERDIK
.
Notation 3. Let functional var denote the total variation of functions on [0, T ], and let N BV [0, T ]=
{G : G(0) = 0, var(G) < ∞}, i.e. N BV [0, T ] = C[0, T ]∗ , the dual space of C[0, T ], the space of all continuous
linear functionals on C[0, T ]. We will denote the characteristic functional of process Z(t) with trajectories
a.s. in C[0, T ] by
ϕZ (G) $ EeihZ,Gi ,
G ∈ N BV [0, T ], where
ZT
hZ, Gi $
Z(t)dG(t).
0
The following notations are needed for the sake of brevity:
Notation 4. Let us denote the set of all permutations of the elements of a set A by Perm(A).
Notation 5. Let m ∈ {2, 3, 4, . . . }, τ = (i1 , . . . , im−1 ) ∈ Perm(2, 3, . . . , m), t = (t1 , . . . , tm ) ∈ [0, T ]m .
Then
Dτ (t) $ |ti1 − t1 | + |ti2 − ti1 | + · · · + tim−1 − tim−2 + t1 − tim−1 .
For m = 1 define Dτ (t) $ 0, t ∈ [0, T ].
In particular, if m = 2,
Dτ (t1 , t2 ) = 2 |t1 − t2 | ,
and for m = 3,
Dτ (t1 , t2 , t3 ) = 2 (max(t1 , t2 , t3 ) − min(t1 , t2 , t3 )) = 2 diam (t1 , t2 , t3 ) .
Theorem 4. The cumulant of the m-dimensional distribution of the stationary DLG process R(t) defined
by (2.2) is
 −m
δ −α


,

 2 σ2
−m
cum (R(t1 ), . . . , R(tm )) =
 δ −α


 2 σ2
X
if
m=1
eαDτ (t1 ,... ,tm ) , if
m > 2.
(2.22)
τ ∈Perm(2,... ,m)
In a neighborhood of 0 ∈ N BV [0, T ] the logarithm of the characteristic functional of R can be put into the
following form.
−m Z
∞
δ X im −α
log ϕR (G) =
2 m=1 m σ 2
eαDτ0 (t) dG(t1 ) . . . dG(tm ),
(2.23)
[0,T ]m
for G ∈ N BV [0, T ] : var(G) < −α/σ 2 , where τ0 is the identity permutation.
Proof. As we have seen, Theorem 3 permits of deriving the distributional properties of the δ-dimensional
DLG process from those of the one-dimensional process. Thus, in effect we have to deal only with the
one-dimensional stationary DLG process R(t) = X 2 (t), where X(t) is the stationary OU process (2.19).
Now consider the cumulants. For m = 1,
1
cum (R(t)) = E R(t) =
2
−α
σ2
−1
,
(2.24)
35
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
because R(t) ∼ Γ 1/2, −σ 2/α by Proposition 6. The right-hand side of (2.24) is the first line of the
right-hand side of (2.22) for δ = 1. For m > 2, by Proposition 27 in the Appendix, we have
X
cum (R(t1 ), . . . , R(tm )) = 2m−1
cov (X(t1 ), X(ti1 )) cov (X(ti1 ), X(ti2 )) · · ·
(i1 ,... ,im−1 )∈Perm(2,... ,m)
X
= 2m−1
τ ∈Perm(2,... ,m)
× cov X(tim−2 ), X(tim−1 ) cov X(tim−1 ), X(t1 )
2 m
σ
eαDτ (t) ,
−2α
which is the second line of the right-hand side of (2.22) for δ = 1. We remark that one can get the same result
by using formula (2.9) in [40], which is a direct expression for the joint cumulants of multiple Wiener–Itô
integrals.
Let us now calculate the characteristic functional. Or rather its logarithm, inasmuch as the cumulants
will be used in the series expansion.
log ϕR (G) = log ϕhR,Gi (u)|u=1 = log ϕRT
0
R(t)dG(t)
(u)|u=1

 T
Z
∞
m
X
i
cumm  R(t)dG(t) ,
=
m!
m=1
(2.25)
0
where cumm denotes the cumulant of order m. Since



 T
Z
ZT
ZT



cumm  R(t)dG(t) = cum 
 R(t)dG(t), . . . , R(t)dG(t)
0
0
0
{z
|
}
m in all
ZT
ZT
cum R(t1 ), . . . R(tm) dG(t1 ) . . . dG(tm )
···
=
0
0
| {z }
m in all
Z
=
[0,T ]m
=
δ
2
δ
2
−α
σ2
−α
σ2
−m
−m
X
eαDτ (t1 ,... ,tm ) dG(t1 ) . . . dG(tm )
τ ∈Perm(2,... ,m)
Z
(m − 1)!
eαDτ0 (t) dG(t1 ) . . . dG(tm ),
(2.26)
[0,T ]m
for m > 2, and the left-hand side equals the right-hand side also for m = 1. In (2.26) the change of the
order of the cumulant operator and the integrals can be justified, e.g. by the following way. Let us express
the cumulant via moments. Then in each term of products of moments one can change the order of the
expectations and the multiple integrals, because the conditions of the Fubini theorem are fulfilled. Then
changing back the expression via moments to cumulant one gets the second row of (2.26).
Returning to (2.25) and using (2.26), we have the result as in (2.23). It is easy to see that the restriction
G ∈ N BV [0, T ] : var(G) < −α/σ 2 is enough for the series on the right-hand side of (2.25) to be convergent.
Now consider the stationary DLG process R(t) on the whole line, i.e. t ∈ R. We will be interested in
the spectra of the process, namely, the second and third order ones, i.e. the spectrum and the bispectrum,
respectively. Of the higher order spectra it is the bispectrum that has the least order. The notion of higher
order spectrum arises from Brillinger, see [3].
Definition 8. Let m ∈ {2, 3, 4, . . . }. Let X(t), t ∈ R, be a stationary process with finite absolute moments
of order m. Assume that the joint cumulant function, taken as a function of m − 1 variables t(m−1) $
36
E. IGLÓI AND C. TERDIK
(t1 , . . . , tm−1 ), is integrable, i.e.
Z
|cum (X(0), X(t1 ), . . . , X(tm−1 ))| dt(m−1) < ∞,
Rm−1
and that the (m − 1)-variable Fourier transform
Sm,X ω (m−1) $
1
2π
m−1 Z

exp −i
m−1
X

tj ωj  cum (X(0), X(t1 ), . . . , X(tm−1 )) dt(m−1) ,
j=1
Rm−1
where ω (m−1) $ (ω1 , . . . , ωm−1 ), is also integrable, i.e.
Z
Sm,X (ω (m−1) ) dω (m−1) < ∞.
Rm−1
Then Sm,X (ω (m−1) ) is called the m-th order spectrum of process X(t). For m = 2 and m = 3 the m-th
order spectrum is called spectrum and bispectrum, respectively.
Remark 5. The spectrum of order m > 3 is generally, complex-valued, and so is the bispectrum. Only the
second order spectrum is always real-valued.
Lemma 2. For each m ∈ {2, 3, 4, . . . } the cumulant function of the stationary DLG process R(t), taken as
a function of m − 1 variables, is integrable, i.e.
Z
|cum (R(0), R(t1 ), . . . , R(tm−1 ))| dt(m−1) < ∞.
Rm−1
Proof. By (2.22), performing an integral transformation, we have
Z
|cum (R(0), R(t1 ), . . . , R(tm−1 ))| dt(m−1)
Rm−1
δ
=
2
δ
=
2
6
δ
2
−α
σ2
−α
σ2
−α
σ2
−m
−m
Z
X
eαDτ (0,t1 ,... ,tm−1 ) dt(m−1) .
τ ∈Perm(1,... ,m−1)Rm−1
Z
(m − 1)!
−m
 
exp α 
Rm−1
Z
(m − 1)!
m−1
X
j=1

exp α
m−1
X

m−1
X |sj | + sj  ds(m−1)
j=1 
|sj | ds(m−1) < ∞,
j=1
Rm−1
where we also utilized the negativity of α in both of the inequalities.
Proposition 8. The spectrum of the stationary DLG process R(t) exists, and it is
1 δ
·
S2,R (ω) =
2π 2
−α
σ2
−2
−4α
·
ω 2 + 4α2
(2.27)
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
37
Proof. By Lemma 2, Definition 8 and formula (2.22), the Fourier transform of the second order cumulant
function with one variable is,
S2,R (ω) =
=
=
1
2π
Z
e−itω cum (R(0), R(t)) dt
R
1 δ
·
2π 2
1 δ
·
2π 2
−α
σ2
−α
σ2
−2 Z
−2
e−itω+2α|t| dt
R
−4α
·
ω 2 + 4α2
S2,R (ω) is also integrable, thus it is really a spectrum.
Remark 6. By the previous proposition the spectrum of the stationary DLG process R(t) is the same as
that of an Ornstein–Uhlenbeck (AR(1)) process.
Notation 6. The symmetrization operator sym is defined by the formula
(ω(3) )
X
1
g (ωi1 , ωi2 , ωi3 ) .
sym (g (ω1 , ω2 , ω3 )) =
3!
(ω(3) )
(i1 ,i2 ,i3 )∈Perm(1,2,3)
Proposition 9. The bispectrum of the stationary DLG process R(t) exists, and it is
1 2 −α −3
1
1
·
δ
3!
sym
,
S3,R ω (2) =
2π
σ2
(ω ) −iω1 − 2α iω2 − 2α
(2.28)
(3)
where ω1 + ω2 + ω3 = 0. In another form (2.28) is written as
−3
1 2
α2 ω1 ω2 + ω12 + ω22 + 12α2
−α
·
48δ
S3,R ω (2) =
2
2π
σ2
(ω12 + 4α2 ) (ω22 + 4α2 ) (ω1 + ω2 ) + 4α2
(2.29)
Hence, S3,R ω (2) has the extraordinary property that it is real-valued moreover it is positive.
Proof. By Lemma 2, Definition 8 and formula (2.22), the Fourier transform of the second order cumulant
function with two variables is,
1 2 Z
e−i(t1 ω1 +t2 ω2 ) cum (R(0), R(t1 ), R(t2 ) dt
S3,R ω (2) =
2π
=
=
=
1
2π
1
2π
1
2π
2
2
2
R2
δ
2
δ
2
δ
2
−α
σ2
−α
σ2
−α
σ2
−3 Z
−3
e−i(t1 ω1 +t2 ω2 )
R2
Z
2!
−3
X
eαDτ (0,t1 ,t2 ) dt
τ ∈Perm(1,2)
e−i(t1 ω1 +t2 ω2 )+α(|t1 |+|t2 −t1 |+|t2 |) dt
R2
Z
2!
R2
e−i(s1 (ω1 +ω2 )+s2 ω2 )+α(|s1 |+|s2 |+|s1 +s2 |) ds.
(2.30)
38
E. IGLÓI AND C. TERDIK
Breaking up the double integral according to the six parts of the plane defined by the axes and the line
y = −x, we have
Z
1
1
·
e−i(s1 (ω1 +ω2 )+s2 ω2 )+α(|s1 |+|s2 |+|s1 +s2 |) ds = 3! sym
·
−iω1 − 2α iω2 − 2α
ω
)
(
(3)
R2
Continuing (2.30), we have for the Fourier transform
S3,R ω (2)
2 −3
1
1
−α
·
δ
3! sym
=
σ2
(ω(3) ) −iω1 − 2α iω2 − 2α
2 −3
8α2 ω1 ω2 + ω12 + ω22 + 12α2
−α
1
·
δ
3!
=
2π
σ2
(ω12 + 4α2 ) (ω22 + 4α2 ) (ω1 + ω2 )2 + 4α2
1
2π
(2.31)
We remark that though it does not show at first sight, the numerator of the fraction in the last row is
symmetric in the variables ω1 , ω2 , ω3 . Now, if ω1 and ω2 are both nonnegative or both negative, the fraction
in the last row of (2.31) can be bounded as
8α2 ω1 ω2 + ω12 + ω22 + 12α2
0<
(ω12 + 4α2 ) (ω22 + 4α2 ) (ω1 + ω2 )2 + 4α2
2
8α2 (ω1 + ω2 ) + 12α2
6
2
(ω12 + 4α2 ) (ω22 + 4α2 ) (ω1 + ω2 ) + 4α2
=
6
8α2
(ω12
+
4α2 ) (ω22
+
4α2 ) (ω22
+
4α2 )
+
4α2 )
8α2
(ω12
+
+
64α4
2
(ω12 + 4α2 ) (ω22 + 4α2 ) (ω1 + ω2 ) + 4α2
(ω12
+
64α4
,
+ 4α2 ) 4α2
4α2 ) (ω22
and so S3,R ω (2) is integrable on the first and third quarters of the plane. If ω1 and ω2 have different signs,
then
8α2 ω1 ω2 + ω12 + ω22 + 12α2
0<
2
(ω12 + 4α2 ) (ω22 + 4α2 ) (ω1 + ω2 ) + 4α2
8α2 ω12 + ω22 + 12α2
6
(ω12 + 4α2 ) (ω22 + 4α2 ) (ω1 + ω2 )2 + 4α2
=
8α2
2
(ω22 + 4α2 ) (ω1 + ω2 ) + 4α2
+
6
+
8α2
2
(ω12 + 4α2 ) (ω1 + ω2 ) + 4α2
32α4
2
(ω12 + 4α2 ) (ω22 + 4α2 ) (ω1 + ω2 ) + 4α2
8α2
2
(ω22 + 4α2 ) (ω1 + ω2 ) + 4α2
+
8α2
2
(ω12 + 4α2 ) (ω1 + ω2 ) + 4α2
+
(ω12
+
32α4
,
+ 4α2 ) 4α2
4α2 ) (ω22
and thus, S3,R ω (2) is integrable on the second and fourth quarters of the plane, too. This means that
S3,R ω(2) is integrable on the whole plane R2 , thus it is really a bispectrum. We have also proven that it
is real-valued and positive.
39
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
3. Superposition of DLG processes and its multifractal limit process
3.1. Superposition of random-parameter stationary DLG processes (SDLG process)
Derivation of LRD by superposing short range dependent processes is not new in the literature. Here are
some papers on this subject: reference [28] showed that superposition of many sources exhibiting the infinite variance syndrome can result in LRD. Reference [13] aggregated random coefficient AR(1) time series,
while [27, 32] superposed renewal processes and alternating renewal processes with heavy tailed interarrival
time distributions to get LRD. A model similar to the latter was considered in [39, 42], aggregating independent ON/OFF processes with heavy tailed ON or OFF time periods to achieve LRD. References [5, 6]
pointed out that the Riemann–Liouville fractional integral operator can be represented as a mixture of AR(1)
operators. Reference [16] considered 1) superposition of random-coefficient independent OU processes; 2)
mixture of random-coefficient OU processes with common input and 3) superposition of random-intensity
Poisson alternating 0–1 reward processes, and the result was a LRD process, called GMOU process in all
the three cases. Reference [1] studied the superposition of OU processes with Lévy process input, and under
specified conditions, obtained a LRD process as a result.
The superposition model to be exposed in this subsection is similar to the first model of [16]; however
the component processes to be superposed are not OU but another sort of AR(1) processes. They are DLG
processes. Going into details, we will show that if we take a sequence of series of independent stationary
DLG processes
R1,1 (t)
R2,1 (t),
..
.
Rn,1 (t),
..
.
R2,2 (t)
(3.1)
... ,
Rn,n (t)
with properly chosen random parameters, the row sums will converge in a certain sense to a limit process
Y , i.e.
Zn $
n
X
k=1
Rn,k −→ Y.
n→∞
As we will see, the characteristic functional of the limit process Y (t), called Superposition of DLG processes
(SDLG), is some weighted geometric mean of the characteristic functionals of stationary DLG processes.
The latter processes are called component processes. As an outcome, both the joint cumulants and the
spectra of the SDLG process Y (t) are mixtures of the joint cumulants and the spectra, respectively, of the
component processes. This fact makes it possible to calculate the cumulants and spectra of process Y (t)
from those of the component processes. Examining the cumulants, one can observe that the SDLG process
Y (t) has fractal-like properties in that its second order cumulant structure is asymptotically the same as
that of the continuous-time fractional Gaussian noise. Moreover, it is LRD, and its centered integral process
is asymptotically multifractal. The one-dimensional distribution of Y (t) is also worthy of note: it is Γ. One
can go further and take a limit process of series of SDLG processes which is already exact multifractal, see
Section 3.2. An additional remarkable attribute is that the bispectrum of Y (t) is real-valued moreover it is
positive. The former property of the bispectrum involves some reversibility-like feature of the 3-dimensional
distribution, see Remark 17.
Let us see the construction in details. Assume that there is given a sequence of parameters αk , k ∈ N, such
that they are independent random variables on a probability space (ΩA , FA , PA ), and −2αk ∼ Γ(3−2H, 1/λ),
k ∈ N, where λ > 0 and 1/2 < H < 1 are constants. Assume moreover that the parameters σk2 depend
on parameters αk , such that σk2 = −αk σ02 , where σ02 > 0 is a constant. Furthermore let the parameters
µn $ µ0 /n, n ∈ N, where µ0 > 0 is a constant. Thus parameters αk and σk are random variables,
parameters µn are deterministic, and the parameters δn,k $ µn /σk2 (see (2.3)) are random variables too.
40
E. IGLÓI AND C. TERDIK
Let the probability space (ΩB , FB , PB )2 be independent of (ΩA , FA , PA )1 . Let the sequence of series (3.1)
of the stationary DLG processes be defined PA a.s. on (ΩB , FB , PB ), specifically, the process Rn , k(t) with
input BM Bk (t) and parameters αk , σk , and µn , for each (n, k), n ∈ N, k = 1, . . . , n. Assume that the BMs
Bk (t), k ∈ N, are independent. Thus, we have
q
(3.2)
dRn,k (t) = (µn + 2αk Rn,k (t)) dt + 2σk Rn,k (t)dBk (t), PA a.s.,
n ∈ N, k = 1, . . . , n.
Now we summarize the above conditions.
Condition 1. λ > 0, 1/2 < H < 1, σ02 > 0 and µ0 > 0 are constants.
Condition 2. (ΩA , FA , PA ) and (ΩB , FB , PB ) are independent probability spaces.
Condition 3. −2αk ∼ Γ(3 − 2H, 1/λ), k ∈ N, are independent random variables on (ΩA , FA , PA ).
Condition 4. σk2 $ −αk σ02 , k ∈ N.
Condition 5. µn $ µ0 /n, n ∈ N.
Condition 6. Bk (t), k ∈ N, are independent Brownian motions on (ΩB , FB , PB ).
Stressing it again, we have the stationary DLG processes Rn,k (t) defined by (3.2). Specifically, R1,1 (t)
has the parameters −2α1 ∼ Γ(3 − 2H, 1/λ), σ12 = −α1 σ02 and µ1 = µ0 .
Remark 7. Note that it is Condition 5 only that accounts for our having not simply a series but a sequence
of series of DLG processes.
Notation 7. Let us define for each n ∈ N the process Zn (t) to be the sum of the n-th row of (3.1), i.e.
Zn (t) $
n
X
Rn,k (t),
(3.3)
k=1
n ∈ N.
Remember that due to Proposition 6, Rn,k (t) ∼ Γ δn,k /2, −σk2 /αk , where δn,k = µn /σk2 , i.e.
µ0
2
,
σ
.
Rn,k (t) ∼ Γ
2nσk2 0
The first observation is that the one-dimensional limit distribution of Zn (t) remains Γ as n → ∞.
Proposition 10. For every fixed t ∈ R the distribution of the series of random variables Zn (t) PA a.s.
converges weakly to a Γ distribution, that is to say,
µ0 λ
w
2
,
σ
, PA a.s.
Zn (t) −→ Γ
n→∞
2σ02 (1 − H) 0
Proof. Because of the independence of processes Rn,k (t), k = 1, . . . , n, and Condition 4, we have
!
n
µ0 X 1
2
,σ .
Zn (t) ∼ Γ
nσ02
−2αk 0
(3.4)
k=1
Because of Condition 3, it follows, by the strong law of large numbers, that
n
µ0
λ
µ0
1
µ0 X 1
,
−→
EA
= 2·
nσ02
−2αk n→∞ σ02
−2αk
σ0 2(1 − H)
PA a.s.,
(3.5)
k=1
where EA means the expectation with respect to the distribution PA . From (3.4) and (3.5) the statement
follows straightaway.
2 We
use the subscript B and A, to remind the reader of the Brownian motion input process and the parameter α, respectively.
41
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
Corollary 1. The SDLG process Y (t), if it exists, has marginal distribution Γ, that is,
µ0 λ
2
,σ ,
Y (t) ∼ Γ
2σ02 (1 − H) 0
for all t > 0.
Remark 8. For each n ∈ N, PA a.s. holds that the random process Zn (t) is PB a.s. continuous, i.e. PA
a.s.: Zn ∈ C[0, T ] PB a.s. Thus, for each n ∈ N, PA a.s.: process Zn induces a measure PZn on C[0, T ]. The
measure PZn is random in the sense that it depends on the realization (α1 , . . . , αn ). However, it would be
inadequate to call it a “random measure”, so we omit the attribute “random”.
Before the main theorem in this section we need the following lemma:
Lemma 3. PA a.s. holds that the series of measures PZn , n ∈ N, mentioned in Remark 8, is tight.
Proof. We premise that in this proof every statement depending on the realizations of α1 , α2 , α3 , . . . , will be
meant PA a.s.
It follows from Proposition 7 and from the form of the characteristic function of the multi-dimensional Γ
distribution (see the Appendix) that the characteristic function of Rn,k (t) − Rn,k (0) is
ϕRn,k (t)−Rn,k (0) (u) = ϕRn,k (t),Rn,k (0) (u, −u)
− µ0
1 0
1
eαk |t| −2nσ02 αk
2
= I − iuσ0
0 −1
eαk |t|
1
− µ02
−2nσ αk
0
= 1 + u2 σ04 1 − e2αk |t|
.
Calculating the moments from the characteristic function by differentiation, we have
2µ0 σ02 1 − e2αk |t|
·
,
n
−2αk
2
µ0
µ0
4
2αk |t|
1
−
e
+
1
σ04 .
EB (Rn,k (t) − Rn,k (0)) = 12
−2nσ02 αk −2nσ02 αk
EB (Rn,k (t) − Rn,k (0))2 =
Hence
4
EB (Zn (t) − Zn (0)) =
n
X
4
EB (Rn,k (t) − Rn,k (0))
k=1
n
n X
X
+
EB (Rn,k (t) − Rn,k (0))2 EB (Rn,` (t) − Rn,` (0))2
k=1 `=1
!
n
n
n µ0 σ06 X µ0
4µ20 σ04 X X
− 2αk +
6 12
1 t2
n
σ02 n
n2
k=1
k=1 `=1
!
n
6 X
12µ0 σ0
3
2 4
= 4µ0 σ0 1 +
(−2αk ) t2
+
n
n
k=1
6 16µ20 σ04 + 12µ0 σ06 K(α1 , α2 , α3 , . . . ) t2 ,
for all n ∈ N, where K(α1 , α2 , α3 , . . . ) is a constant bounding above the series
1
n
n
P
k=1
(−2αk ). Such a constant
really exists, in agreement with the strong law of large numbers.
Process Zn (t) is stationary, thus it has stationary increments. Consequently, using (3.6), we have
4
EB (Zn (t + s) − Zn (s)) 6 K1 (α1 , α2 , α3 , . . . )t2 ,
for all t, s > 0 and n ∈ N. Thus the tightness condition of Kolmogorov is fulfilled.
(3.6)
42
E. IGLÓI AND C. TERDIK
Now, in the main theorem of this section, we state the existence of the SDLG process Y (t), and give a
series expansion for its logarithmic characteristic functional.
Theorem 5. 1) PA a.s. holds that the series of measures PZn , n ∈ N, mentioned in Remark 8, converges
weakly to some measure PY , i.e. there exists an a.s. continuous process Y (t) on (ΩB , FB , PB ) such that
w
Zn −→ Y
in C[0, T ],
n→∞
PA a.s.
Distribution PY and thus process Y (t) do not depend on the realizations of α1 , α2 , α3 , . . . any more. (Zn
and PZn do.)
2) For all G ∈ N BV [0, T ] such that var(G) < σ0−2 , the logarithm of the characteristic functional of process
Y (t) (or of PY ) can be put into the form
2(H−1)
Dτ0 (t)
dG(t1 ) . . . dG(tm ),
1+
2λ
Z
∞
X
µ0 λ
im 2m
σ0
log ϕY (G) = 2
2σ0 (1 − H) m=1 m
[0,T ]m
(3.7)
where τ0 is the identity permutation. (See Notation 5 on Dτ0 (t).)
3) The characteristic functional of process Y (t) is the weighted geometric mean of the characteristic functionals of stationary DLG component processes R1,1 (t), where the weight measure is the Γ(3 − 2H, 1/λ)
distribution, see Condition 3. In other words, the logarithm of the characteristic functional of process Y (t)
is the mixture of logarithms of the characteristic functionals of component processes R1,1 (t), i.e.
log ϕY (G) = EA log ϕR1,1 (G),
(3.8)
for all G ∈ N BV [0, T ].
Proof. The proof of the first two statements is composed of four parts. The fifth part is the proof of the
third statement of the theorem. We premise that in this proof every statement depending on the realizations
of α1 , α2 , α3 , . . . , will be meant by PA a.s.
Part 1. Because of the independence of the elements inside the rows of (3.1), Theorem 4, formula (2.3),
and Conditions 4 and 5, we have
log ϕZn (G) =
n
X
log ϕRn,k (G)
k=1
∞
X
im
2(m−1)
µ0 σ0
=
m
m=1
Z
[0,T ]m
1X
n
n
k=1
1
αk Dτ0 (t)
e
dG(t1 ) . . . dG(tm ),
−2αk
for var(G) < σ0−2 . Because of the strong law of large numbers and Condition 3,
1X
n
n
k=1
1
eαk Dτ0 (t)
−2αk
−→ EA
n→∞
1
eαk Dτ0 (t)
−2αk
λ
=
2(1 − H)
2(H−1)
Dτ0 (t)
.
1+
2λ
Consequently
log ϕZn (G) −→
n→∞
Z
∞
X
µ0 λ
im 2m
σ
2σ02 (1 − H) m=1 m 0
[0,T ]m
2(H−1)
Dτ0 (t)
dG(t1 ) . . . dG(tm ),
1+
2λ
(3.9)
for var(G) < σ0−2 . The change of the order of the limit and the integral, and also the change of the order of
the limit and the sum
that we are considering the characteristic functionals
can be easily justified by the fact
only on the domain G ∈ N BV [0, T ] : var(G) < σ0−2 .
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
43
Part 2. As we found it simpler to deal directly with the characteristic functionals of measures on C[0, T ] than
to prove the convergence of finite-dimensional distributions, we will use the following well-known property
of measures on complete separable metric spaces.
w
PZn −→ P
Every subsequence of hPZn i contains
a subsequence converging weakly to P .
⇐⇒
n→∞
(3.10)
Let hPZ ne i be a subsequence of hPZn i. We have the following deduction chain.
Lemma 3
=⇒
The series hPZ ne i is tight.
=⇒
=⇒
The series hPZ ne i is relatively compact.
of hPZ ne i , converging weakly to some measure, say P .
There exists a subsequence PZ ≈
n
=⇒
log ϕZ ≈ (G) −→ log ϕP (G)
n
n→∞
for all G ∈ N BV [0, T ],
and P , respectively.
where ϕZ ≈ (G) and ϕP (G) is the characteristic functional of PZ ≈
n
n
From the last convergence and (3.9), we have
log ϕZ ≈ (G) −→ log ϕP (G)
n
n→∞
Z
∞
X
im 2m
µ0 λ
σ0
= 2
2σ0 (1 − H) m=1 m
[0,T ]m
2(H−1)
Dτ0 (t)
dG(t1 ) . . . dG(tm ),
1+
2λ
(3.11)
for var(G) < σ0−2 .
If one takes another
e i instead of hPZ ne i, then it follows in the same way that there exists
subsequence hPZ m
i,
and
a
probability
measure Q on C[0, T ] such that
of
hP
a subsequence PZ m
≈
Zm
e
log ϕZ m
≈ (G) −→ log ϕQ (G)
n→∞
Z
∞
X
im 2m
µ0 λ
σ
=
2σ02 (1 − H) m=1 m 0
1+
[0,T ]m
Dτ0 (t)
2λ
2(H−1)
dG(t1 ) . . . dG(tm ),
(3.12)
for var(G) < σ0−2 . It is simply the consequence
ϕP (G) = ϕQ (G),
for var(G) < σ0−2 ,
(3.13)
that concerns us in (3.11) and (3.12). By (3.10), what we have to prove is that P = Q.
Part 3. To prove that P = Q, it is enough to prove that all the finite-dimensional marginal distributions of
the two measures are the same. In other words, we have to prove that all the finite-dimensional distributions
of the a.s. continuous processes, say, YP and YQ , are the same, where YP and YQ has distribution P and Q,
respectively.
Let m ∈ N and t1 , . . . , tm ∈ [0, T ] be arbitrary. We have to prove that the distribution of (YP (t1 ), . . . ,
YP (tm )) and that of (YQ (t1 ), . . . , YQ (tm )) on Rm are the same. To attain this it is enough to prove that all
the linear combinations of their components have the same one-dimensional distribution, i.e.
m
X
λk YP (tk ) ∼
k=1
m
X
λk YQ (tk ),
k=1
for all λ1 , . . . , λm ∈ R.
In (3.13) let us perform the substitution
G$u
m
X
k=1
λk χ[tk ,T ] ,
(3.14)
44
E. IGLÓI AND C. TERDIK
where
m
X
|u| < σ0−2
!−1
|λk |
,
(3.15)
k=1
and χ denotes the characteristic (or indicator) function of the set in its subscript. This replacement is
legitimate because obviously var(G) < σ0−2 . In this way we have
m
ϕP
k=1
λk YP (tk )
m
(u) = ϕP (G) = ϕQ (G) = ϕ P
k=1
λk YQ (tk )
(u),
for the values of u given by (3.15). Thus, in a neighborhood of u = 0 the characteristic functions are equal,
from which the equality of the moments of the two random variables in (3.14) follows. To complete the proof
we have to prove that these moments determine the distribution uniquely.
Part 4. We have to prove that the moments of the random variable
m
X
λk YP (tk )
(3.16)
k=1
uniquely determine the distribution, and the proof is complete.
It is well known that the moments Mn , n ∈ N, of a distribution, uniquely determine the distribution, if
the absolute moments of even order, M 2n , n ∈ N, satisfy the moment condition
1
M 2n 2n
< ∞.
lim
n→∞
2n
In our case
1
2n ! 2n
m
P
λk YP (tk )
E lim
n→∞
m
X
1
= lim 
λk YP (tk )
n→∞
2n 2n
n→∞
6 lim

k=1
m
1 X
|λk | kYP (t)k(2n)
2n
k=1
!
=
k=1
m
X
|λk | lim
n→∞
k=1


(2n)
1
kYP (t)k(2n)
2n
< ∞,
because YP (t) has Γ distribution, see Proposition 10. Thus, the moments uniquely determine the distribution
of the random variable (3.16), and this is what was to be proved. The first two statements of the theorem
are proven.
Part 5. Similarly to the beginning of Part 1, we have
EA log ϕR1,1 (G) = EA
Z
∞
X
im
2(m−1)
µ0 σ0
m
m=1
Z
∞
X
im
2(m−1)
µ0 σ0
=
m
m=1
[0,T ]m
[0,T ]m
1
eαk Dτ0 (t) dG(t1 ) . . . dG(tm )
−2αk
1
αk Dτ0 (t)
e
dG(t1 ) . . . dG(tm )
−2αk
EA
Z
∞
X
im 2m
µ0 λ
σ
= 2
2σ0 (1 − H) m=1 m 0
[0,T ]m
= log ϕY (G),
2(H−1)
Dτ0 (t)
dG(t1 ) . . . dG(tm )
1+
2λ
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
45
for var(G) < σ0−2 . The change of order of the expectation and the integral, and also the change of order of
the expectation and the sum can be easily justified by Fubini’s theorem and the fact that the characteristic
functionals are only considered for var(G) < σ0−2 .
We also have to prove, that equation (3.8) holds not only for var(G) < σ0−2 but on the whole space
N BV [0, T ]. The exponential of the right-hand side of (3.8) defines a characteristic functional, ψ(G), say, on
the whole N BV [0, T ] because every positive power of the characteristic functional of any stationary DLG
process also will be a characteristic functional (of some stationary DLG process), see Remark 3. Now, we have
two characteristic functionals, ϕY (G) and ψ(G), and they both agree with the exponential of the right-hand
side of (3.7) for var(G) < σ0−2 . The situation is the same as it was at (3.13), thus, the conclusion must also be
the same, namely that ϕY (G) = ψ(G) for all G ∈ N BV [0, T ]. Thus (3.8) holds for all G ∈ N BV [0, T ].
The following observation is a consequence of the third statement of Theorem 5 and Remark 3.
Corollary 2. The SDLG process Y (t) is infinitely divisible.
Remark 9. As it appears from the last paragraph of the previous proof, the existence of the SDLG process
Y (t) depends eventually on the stationary DLG process being infinitely divisible.
Remark 10. Let G(t) $ uχ[t1 ,T ] (t), where u ∈ R, |u| < σ0−2 . Taking function G(t) in Theorem 5, we have
log ϕY (t) (u) = −
µ0 λ
2σ02 (1 −
H)
log 1 − iuσ02 ,
for |u| < σ0−2 . This is the logarithm of the characteristic function of a Γ distribution around zero, and it
uniquely determines the characteristic functionon the whole line. Thus, we have the result already known,
namely that Y (t) ∼ Γ µ0 λσ0−2 (1 − H)−1 /2, σ02 , see Corollary 1.
The following lemma will be needed later:
Lemma 4. Let m ∈ N, t1 , . . . , tm ∈ [0, T ]. Then the mixture representation
cum (Y (t1 ), . . . , Y (tm )) = EA cum (R1,1 (t1 ), . . . , R1,1 (tm ))
holds, where R1,1 (t) are the component processes defined above, see the paragraph preceding Remark 7. The
following explicit formula also can be given.
µ0 λσ02m−2
cum (Y (t1 ), . . . , Y (tm )) =
2(1 − H)
X
τ ∈Perm(2,... ,m)
2(H−1)
Dτ (t)
.
1+
2λ
(3.17)
Proof. The same mixture representation holds for the finite-dimensional, in particular for the m-dimensional
distributions, as for the distributions of processes on C[0, T ]. Thus the same formula holds for the logarithmic
m variate characteristic functions, and for the cumulants of order m too. This is
= µ0 σ02m−2
cum (Y (t1 ), . . . , Y (tm )) = EA cum (R(t1 ), . . . , R(tm ))
2(H−1)
X
X
1
µ0 λσ02m−2
Dτ (t)
αk Dτ (t)
EA
e
.
=
1+
−2αk
2(1 − H)
2λ
τ ∈Perm(2,... ,m)
τ ∈Perm(2,... ,m)
The second equation follows from (2.22, 2.3), and Condition 4 and that µ = µ0 . The last equation follows
from Condition 3.
Proposition 11. The centered integral process of the SDLG process Y (t) is asymptotically multifractal.
Specifically, the following scaling rules hold.

 t
Z
cumm  (Y (s) − E Y )ds ' k(m)tm+2(H−1) ,
t→∞
0
46
E. IGLÓI AND C. TERDIK
for m > 2, where cumm denotes the cumulant of order m of the one-dimensional distribution, k(m),
m > 2, are some positive constants not depending on t but depending on m, and ' means asymptotically
t→∞
equivalence as t → ∞.
Proof. Taking in (3.7) function G(t) = ut with |u| sufficiently small, we have

cumm 
ZT
0
'
T →∞

µ0 λσ02m−2
(m − 1)!
Y (t)dt =
2(1 − H)
[0,T ]m
Z
21−2H
µ0 λ3−2H σ02m−2 (m − 1)!
1−H
Z
Dτ0 (t)
1+
2λ
2(H−1)
dt
Dτ0 (t)2(H−1) dt T m+2(H−1) = k(m)T m+2(H−1) ,
[0,1]m
where the asymptotically equivalence holds for m > 2.
Proposition 12. The SDLG process Y (t) is LRD with parameter H.
Proof. Using Lemma 4, the autocovariance function of the stationary process Y (t) is
RY (t) $ cov(Y (0), Y (t)) =
σ02
µ0 λ
2(1 − H)
|t|
1+
2λ
2(H−1)
' const.t2(H−1) ,
t→∞
indicating a hyperbolically decay, and the LRD parameter is H.
3.2. Multifractal limit processes (LISDLG and ∆LISDLG processes)
In this subsection we will show that one can go further and make up a subsequent limit process as
parameter λ → 0. The new limit process will have interesting properties. Firstly it has the same second
order covariance structure as that of the FBM, thus it will be second order self-similar. However, not selfsimilar, but multifractal, in particular exactly multifractal. Furthermore, the increments of the limit process
will have a distribution close to Γ, leastwise for the case of values of H close to 1. In addition, the bispectrum
will also be unusual as it is real-valued, but that is in fact
the subject of a later section.
Assume that µ0 depends on λ so that lim µ0 λ3−2H = c0 , some positive constant. Let us denote the
λ→0
dependence of the SDLG process on λ, too. Hence Y (t) = Yλ (t).
As a consequence of Corollary 1, we know the expectation of Yλ (t), and thus
µ0 λ3−2H 2(H−1)
µ0 λ
=
λ
−→ ∞,
λ→0
2(1 − H)
2(1 − H)
E Yλ (t) =
because 1/2 < H < 1, see Condition 1. Hence, the higher order moments of Yλ (t), and also the central
moments converge to infinity, therefore the existence of any limit process as λ → 0 is unlikely, even after standardization. However, there may exist a limit process in some Schwartz-distribution sense, which
indicates that it is more profitable to deal with the integral or time-aggregated process.
Notation 8.
Zt
Jλ (t) $
(Yλ (s) − E Yλ ) ds,
0
where the integral is simply assumed to be either pathwise Lebesgue integral or pathwise Riemann integral.
Process Jλ (t) is the integral process of the centralized SDLG process Yλ , therefore it will be referred to as
an Integrated SDLG (ISDLG) process. Its discrete increment process,
∆Jλ (t) $ Jλ (t + 1) − Jλ (t),
t ∈ Z, will be called a ∆ISDLG process.
47
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
Remark 11. The ISDLG process Jλ (t) is infinitely divisible because so is the SDLG process Yλ (t), see
Corollary 2.
We reassert our latter assumption.
Condition 7. lim µ0 λ3−2H = c0 > 0.
λ→0
Because Jλ (t) is an integral process, it is a.s. continuous, hence it generates a measure on C[0, T ]. Now
we state that the characteristic functional of the ISDLG process Jλ converges as λ → 0, around zero, at the
very least.
Proposition 13. In a neighborhood of 0 ∈ N BV [0, T ] the logarithm of the characteristic functional of the
ISDLG process Jλ converges to a limit, that is to say,

 T
Z
∞
X
im
cumm  Jλ (t)dG(t)
log ϕJλ (G) =
m!
m=1
0
Z
∞
c0 X im 2m
2
· 2
σ
−→
λ→0 1 − H
σ0 m=2 m 0
1−2H
2(H−1)
Dτ0 (t)
m
Y
(G(T ) − G(tk )) dt,
(3.18)
k=1
[0,T ]m
for
var(G) <
1
,
T σ02
(3.19)
where τ0 is the identity permutation. (See Notation 5 on Dτ0 (t).)
Proof. Let G ∈ N BV [0, T ], such that (3.19) holds. Let us calculate the cumulants of the random variable
ZT
Jλ (t)dG(t).
(3.20)
0
The cumulant of order 1 is the expectation, which is zero, because Jλ (t) is centralized, i.e.

 T
Z
ZT
cum1  Jλ (t)dG(t) = E Jλ (t)dG(t) = 0.
0
(3.21)
0
The higher order cumulants are


 T
 T
Z
Z
cumm  Jλ (t)dG(t) = cumm  Yλ (t) (G(T ) − G(t)) dt
0
0
Z
cum (Yλ (t1 ), . . . , Yλ (tm ))
=
(G(T ) − G(tk )) dt
k=1
[0,T ]m
=
m
Y
µ0 λ
σ 2m−2
2 (1 − H) 0
Z
X
[0,T ]m τ ∈Perm(2,... ,m)
= µ0 λ3−2H
−→ c0
λ→0
21−2H 2m−2
σ
(m − 1)!
1−H 0
21−2H 2m−2
σ
(m − 1)!
1−H 0
Z
2(H−1) Y
m
Dτ (t)
(G(T ) − G(tk )) dt
1+
2λ
k=1
Z
2(H−1)
(2λ + Dτ0 (t))
[0,T ]m
(G(T ) − G(tk )) dt
k=1
[0,T ]m
Dτ0 (t)2(H−1)
m
Y
m
Y
k=1
(G(T ) − G(tk )) dt,
(3.22)
48
E. IGLÓI AND C. TERDIK
for m > 2. In the second equation the change of the order of the cumulant operator and the integrals can be
justified similarly as in (2.26) (see the explanation after (2.26)). In the third equation we used Lemma 4, and
the fourth equation arises by performing the proper permutational integral transformations. In reference to
the convergence in the last row, we remark that if the m-dimensional integral in the last row were finite,
then Lebesgue’s dominated convergence theorem could be applied, since
(2λ + Dτ0 (t))2(H−1) 6 Dτ0 (t)2(H−1) .
(3.23)
Our backlog, the inequality
Z
Dτ0 (t)2(H−1)
m
Y
|G(T ) − G(tk )| dt < ∞,
k=1
[0,T ]m
will be a by-product of the following calculation.

 T
Z
Z
m
3−2H 1−2H 2m−2
Y
2
σ0
(m − 1)!
2(H−1)
cumm  Jλ (t)dG(t) 6 µ0 λ
D
(t)
|G(T ) − G(tk )| dt
τ
0
1−H
k=1
m
0
[0,T ]
!m Z
2m−2
µ0 λ3−2H 21−2H σ0
(m − 1)!
sup |G(T ) − G(t)|
6
Dτ0 (t)2(H−1) dt
1−H
t∈[0,T ]
6 const.
= const.
− 1)! var(G)
1−H
σ02m−2 (m
m
[0,T ]m
Z
2(H−1)
|t2 − t1 |
dt
[0,T ]m
m
(m − 1)!T 2(H−1)
σ02 var(G)T
,
2
σ0 (1 − H) (2H − 1)H
(3.24)
where the first inequality is the consequence of the second last row of (3.22) and of (3.23). The second
inequality is obvious. The third inequality and the last equality hold because of Condition 7 and the
1/2 < H < 1 part of Condition 1.
Using (3.24) we find the series expansion in the first row of (3.18) to be absolutely convergent on the
domain given by (3.19). Substituting (3.21) and (3.22) into the first row of (3.18), we get the statement to
be proved. In the last step, when changing the order of the infinite sum and the operator lim , we could
λ→0
apply Lebesgue’s theorem because of (3.24).
Lemma 5. The set of measures on C[0, T ] induced by the ISDLG processes Jλ , λ > 0, is tight.
Proof. Let λ > 0 be arbitrary. The ISDLG process Jλ (t) has stationary increments, therefore
2
E (Jλ (s + t) − Jλ (s)) = E Jλ (t)2 = cum2 (Jλ (t)) .
(3.25)
We will refer to the proof of Proposition 13. It is easy to check that the equation part of (3.22), i.e. (3.22)
except for the last row, holds for all G ∈ N BV [0, T ]. Performing the substitution G(s) = χ[t,T ] (s) in the
second last row of (3.22) for m = 2, we have
3−2H
cum2 (Jλ (t)) = µ0 λ
σ2
6 const. 0
1−H
Z
21−2H 2
σ
1−H 0
Z
(2λ + 2 |s2 − s1 |)
ds2 ds1
[0,t]2
|s2 − s1 |2(H−1) ds2 ds1 = const.
[0,t]2
2(H−1)
σ02
t2H ,
(2H − 1)H (1 − H)
(3.26)
where the inequality follows because of the 1/2 < H < 1 part of Condition 1. Equations (3.25) and (3.26)
together show that Kolmogorov’s tightness condition is satisfied.
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
49
Now in the main theorem in this subsection, we state the convergence as λ → 0, of the family of ISDLG
processes Jλ (t), λ > 0, and give a series expansion for the logarithmic characteristic functional of the limit
process.
Theorem 6. There exists an a.s. continuous process J such that
w
Jλ −→ J
in C[0, T ].
λ→0
In a neighborhood of 0 ∈ N BV [0, T ] the logarithmic characteristic functional of the limit process J has the
series expansion
Z
∞
21−2H c0 X im 2m
·
σ
log ϕJ (G) =
1 − H σ02 m=2 m 0
Dτ0 (t)2(H−1)
m
Y
(G(T ) − G(tk )) dt,
(3.27)
k=1
[0,T ]m
for
var(G) <
1
σ02 T
,
where τ0 is the identity permutation.
Because process J (t) is the Limit of ISDLG processes, it will be called a LISDLG process.
Proof. The reader can check that the proof of Theorem 5 as far as Part 4 can be applied to the present case
too. That is to say, in the same way, we already have the tightness and the convergence of the characteristic
functionals around zero, see Lemma 5 and Proposition 13, respectively. The only point, that we have to
amend is in Part 4. We adapt the notations of the proof mentioned and continue it from the beginning of
Part 4.
We have to prove that the moments of the random variable
m
X
λk JP (tk )
(3.28)
k=1
uniquely determine its distribution, and the proof is complete. Let us consider the expression in the moment
condition. We can estimate it as follows.
 
1 
m
2n  2n
m
1
X
 1
 X
kJP (tj )k(2n) .
lim  E λk JP (tk )   6
|λk | max lim
n→∞
16j6m n→∞ 2n
2n
k=1
k=1
Thus what we have to prove is
lim
n→∞
1
kJP (t)k(2n)
2n
< ∞,
(3.29)
for all t ∈ [0, T ].
For this end we use the expression of moments via cumulants since the cumulants are more tractable now
than the moments. At first let us show how to express the joint central moments of the SDLG process Y (t)
by its joint cumulants. The general formula we apply is formula (1.9) in [40]. Thus we have
E
m
Y
j=1
(Y (tj ) − E Y (t)) =
X
Y
cum Y t(K) .
(3.30)
L∈Part(26) (1,... ,m) K∈L
Explanation of the notations: Part(1, . . . , m) is the set of all partitions of the set {1, . . . , m} and the set of
partitions the elements of which have at least two elements is denoted by Part(26) (1, . . . , m), i.e.
Part(26) (1, . . . , m) $ {L ∈ Part(1, . . . , m) : ∀K ∈ L : |K| > 2} ·
50
E. IGLÓI AND C. TERDIK
Moreover, Y t(K) = (Y (ti1 ), . . . , Y (tik )), if K = {i1 , . . . , ik }.
Replacing the cumulants in (3.30) by their expressions in (3.17), we have
E
m
Y
X
(Y (tj ) − E Y (t)) =
Y

L∈Part(26) (1,... ,m) K∈L
j=1
X
=
L∈Part(26) (1,... ,m)
X
3−2H
2|K|−2
 µ0 λ
σ
2(1 − H) 0
2λ + Dτ

2(H−1)

t(K)
τ ∈Perm(K\{min(K)})

|L|
3−2H
Y
 µ0 λ
σ02m
2
2(1 − H)σ0
X
2λ + Dτ

2(H−1)
.
t(K)
K∈L τ ∈Perm(K\{min(K)})
Hence,
Z
E (Jλ (t))m =
E
(Y (sj ) − E Y ) ds
j=1
[0,t]m
"
X
=
m
Y
L∈Part(26) (1,... ,m)
µ0 λ3−2H
1
·
2
σ0
2(1 − H)

Z
Y 
×
(|K| − 1)!
K∈L
2 m
−→ tσ0
λ→0
[0,t]|K|

|L|
σ02m
X

2(H−1)

ds(K) 
2λ + Dτ0 s(K)
τ ∈Perm(K\{min(K)})

|L| Y
Z
c0 2(H−1)
1


·
t
(|K|
−
1)!


2(1 − H) σ02
X
L∈Part(26) (1,... ,m)
K∈L
Dτ0 s(K)
2(H−1)


ds(K) 
[0,1]|K|
where the change of order of the integral and the limes as λ → 0 is justifiable because of the inequality
2(H−1)
(2λ + Dτ0 (s))
6 Dτ0 (s)2(H−1) .
After these preparations we can return to the actual proof of inequality (3.29).
lim
n→∞
1
kJP (t)k(2n)
2n



 1

6 tσ02 lim
n→∞ 
 2n


 1
 tσ02 2n
= lim
n→∞  2n
X
c0
1
· 2 t2(H−1)
2(1 − H) σ0
L∈Part(26) (1,... ,2n)

1

 2n


Z

2(H−1)
Y 

Dτ0 s(K)
ds(K) 
×
(|K| − 1)!


K∈L

[0,1]|K|
X
L∈Part(1,... ,2n)
|L|


1
2n 
|L| Y 
c0
1
1

· 2 t2(H−1)
(|K| − 1)!

2(1 − H) σ0
(2H − 1)H

K∈L
 

1
 2n




X
Y
c0
1
1 
· 2 t2(H−1) lim
(|K| − 1)!
·
6 tσ02 max 1,
n→∞

(2H − 1)(1 − H)2H σ0
 2n L∈Part(1,... ,2n) K∈L

(3.31)
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
51
At the same time, if we take 2n identical copies of a Γ(1, 1) distributed random variable, i.e. ξ1 = . . .
= ξ2n ∼ Γ(1, 1), and express their joint moment by the joint cumulants, we get the very identity
(2n)! = E ξ12n = E
Y
X
=
2n
Y
X
ξj =
Y
L∈Part(1,... ,2n) K∈L
j=1
X
cum|K| (ξ1 ) =
L∈Part(1,... ,2n) K∈L
cum ξ (K)
Y
(|K| − 1)!,
(3.32)
L∈Part(1,... ,2n) K∈L
where in the third equation the general expression of the moments via cumulants is used again, see formula (1.9)
in [40]. Substituting the result of (3.32) into (3.31), we obtain the inequalities
1
1
1
1
1
kJP (t)k(2n) 6 const. lim
((2n)!) 2n 6 const. lim
·
(1 + 2 + · · · + 2n) < ∞.
lim
n→∞ 2n
n→∞ 2n
n→∞ 2n 2n
We arrived at (3.29), which was to be proved.
Since the ISDLG process Jλ is infinitely divisible (see Rem. 11), and the weak limit of infinitely divisible
processes also has this property, we have the following corollary:
Corollary 3. The LISDLG process J(t) is infinitely divisible.
The ISDLG processes Jλ , λ > 0, being integral processes of stationary processes, have stationary increments, hence the following remark is true:
Remark 12. The LISDLG process J(t) has stationary increments.
After the SDLG process Yλ (t), we defined the ISDLG process Jλ (t), which is asymptotically multifractal,
see Proposition 11. Taking the limit as λ → 0, we have arrived at our object, the limit process J(t). Our
motive to do so was that the LISDLG process J(t) is already exactly multifractal.
Theorem 7. The LISDLG process J(t) has the following cumulants
cum1 (J(t)) = 0,
and
cumm (J(t)) = (m − 1)!
21−2H
2(m−1)
c0 σ0
1−H
Z
Dτ0 (s)2(H−1) ds tm+2(H−1)
[0,1]m
= k1 (m)t
m+2(H−1)
,
(3.33)
for m > 2. (See also Prop. 11 for comparison.) Therefore, the LISDLG process J(t), t ∈ R, is multifractal.
Proof. Let us consider the function G(s) = uχ[t,T ] (s), 0 6 s 6 T , where 0 6 |u| < σ0−2 T −1 , and substitute
it into (3.27). The replacement is legitimate since var(G) = |u| < σ0−2 T −1 , and for the cumulant generator
function log ϕJ(t) (u) we have
log ϕJ(t) (u) = log ϕJ (G)
=
Z
∞
21−2H c0 X im 2m
· 2
σ0
1 − H σ0 m=2 m
Dτ0 (s)2(H−1)
Dτ0 (s)2(H−1) ds
[0,t]m
=
∞
X
m
(G(T ) − G(sk )) ds
k=1
[0,T ]m
Z
∞
m
21−2H c0 X (iu) 2m
·
σ
=
1 − H σ02 m=2 m 0
m
Y
21−2H c0 2m
(iu)
(m − 1)!
·
σ
m!
1 − H σ02 0
m=2
Z
Dτ0 (s)2(H−1) ds,
[0,t]m
52
E. IGLÓI AND C. TERDIK
that is,
cum1 (J(t)) = 0,
and
21−2H c0 2m
·
σ
cumm (J(t)) = (m − 1)!
1 − H σ02 0
= (m − 1)!
1−2H
c0
2
· 2 σ02m
1 − H σ0
Z
Dτ0 (s)2(H−1) ds
[0,t]m
Z
Dτ0 (s)2(H−1) ds tm+2(H−1) ,
[0,1]m
for m > 2.
Corollary 4. The discrete increment process ∆J(t) $ J(t + 1) − J(t), t ∈ Z, is multifractal. We will call it
a ∆LISDLG process.
Proof. Indeed the aggregated increments ∆J(k)(n) , i.e.
∆J(k)(n) =
n−1
1X
1
d 1
∆J(kn − j) = (J(kn + 1) − J((k − 1) n + 1)) = J(n),
n j=0
n
n
have cumulants of order m
Z
21−2H
2(m−1)
c0 σ0
cumm ∆J(k)(n) = (m − 1)!
1−H
Dτ0 (s)2(H−1) ds n2(H−1) = k1 (m)n2(H−1) ,
(3.34)
[0,1]m
as it follows from Theorem 7. The exponent of n is 2H − 2 = m
depends on m:
H(m) =
2H−2+m
m
− 1 , i.e. the Hurst exponent
2H − 2
2H − 2 + m
=
+ 1.
m
m
The following is also an outcome of Theorem 7:
Corollary 5. The LISDLG process J(t) has the same covariance structure as the FBM. That is,
2H
2H
,
cov(J(t1 ), J(t2 )) = const. t2H
1 + t2 − |t2 − t1 |
t1 , t2 > 0, where the constant does not depend on time. Moreover, the ∆LISDLG process ∆J(t) is LRD with
parameter H.
Proof. As the LISDLG process J(t) has stationary increments, we have
cov(J(t1 ), J(t2 )) =
1
D2 (J(t1 )) + D2 (J(t2 )) − D2 (J(t2 − t1 )) ,
2
(3.35)
where the operator D2 denotes the variance. The variance is the second order cumulant, therefore by
substituting into (3.35) the expression for the second order cumulant from Theorem 7, we have
2H
2H
+
t
−
|t
−
t
|
,
cov(J(t1 ), J(t2 )) = const. t2H
2
1
1
2
thus we have proved the first part of the corollary.
LRD is a property depending only on the second order covariance structure. On the other hand, the
H-FGN, the increment process of the FBM with parameter H, is LRD with parameter H. In this manner
the second part is proved too.
53
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
By the first statement of Corollary 5, the second order cumulant structure of the LISDLG process J(t)
is the same as that of the FBM. The question is how to proceed, what is the general formula for the higher
order joint cumulants? The question relative to a closed formula is open. However, on the basis of the
foregoing it is not hard to arrive at the following form:
Proposition 14. The 2 6 m-th order joint cumulants of the LISDLG process J(t) are
 t

Z
cum(J(t1 ), . . . , J(tm )) = const.σ02m (m − 1)! sym  Dτ0 (s)2(H−1) ds ,
t
(3.36)
0
for 0 < t1 , . . . , tm , where the operator sym is the symmetrization operator, i.e. symmetrization for all
t
permutations of the components of the vector t = (t1 , . . . , tm ). Moreover, the constant depends neither on
time, nor on m.
Proof. The proof is similar to that of Theorem 7. Let us consider the function
. X
ui χ[ti ,T ] (s),
G(s) =
m
i=1
0 ≤ s ≤ T , where
m
X
|ui | < σ0−2 T −1 ,
i=1
and substitute it into (3.27). The replacement is legitimate, since
var(G) =
m
X
|ui | < σ0−2 T −1 ,
i=1
and for the cumulant function log ϕJ(t1 ),... ,J(tm ) (u1 , . . . , um ) we have
log ϕJ(t1 ),... ,J(tm ) (u) = log ϕJ (G)
Z
∞
21−2H c0 X ik 2k
·
σ
=
1 − H σ02
k 0
k=2
Dτ0 (s)2(H−1)
k
Y
(G(T ) − G(sj )) ds
j=1
[0,T ]k
Z
Z
m
∞
21−2H c0 X ik 2k X
· 2
σ0
=
ui1 · · · uik · · · Dτ0 (s)2(H−1) ds
1 − H σ0
k
i ,... ,i =1
k=2
=
∞
X
k=2
1
k! i
m
X
1 ,... ,ik
1
k
ti1
tik
0
0
21−2H c0 2k
·
σ (k − 1)! (iu1 ) · · · (iuk )
1 − H σ02 0
=1
Zti1
Ztik
···
0
Dτ0 (s)2(H−1) ds.
0
Thus we have
 t

Z
21−2H c0 2m
·
σ (m − 1)! sym  Dτ0 (s)2(H−1) ds ,
cum(J(t1 ), . . . , J(tm )) =
1 − H σ02 0
t
0
for m ≥ 2.
54
E. IGLÓI AND C. TERDIK
Unfortunately, equation (3.36) is not a closed formula. However, there would be a closed formula for the
d
J(t), if the latter process existed, viz.
joint cumulants of the derivative process J 0 (t) = dt
dm
cum(J(t1 ), . . . , J(tm ))
dt1 · · · dtm
= const.σ02m (m − 1)! sym Dτ0 (t)2(H−1)
cum(J 0 (t1 ), . . . , J 0 (tm )) =
=
X
const.σ02m
t
Dτ (t)2(H−1)
τ ∈Perm(2,... ,m)
= lim cum(Yλ (t1 ), . . . , Yλ (tm )).
(3.37)
λ→0
Because of stationarity, for m = 2 and m = 3 the latter means that
cum(J 0 (0), J 0 (t)) = const2 |t|
2(H−1)
= const02
d
dt
2
2H
|t|
,
(3.38)
for t 6= 0, and
cum(J 0 (0), J 0 (t1 ), J 0 (t2 )) = const3 (|t1 | + |t2 − t1 | + |t2 |)
2(H−1)
2(H−1)
= const3 (2 diam(0, t1 , t2 ))
2
d
2H 0
= const3
|t| dt
,
(3.39)
t=diam(0,t1 ,t2 )
for t1 6= 0, t2 6= 0, t1 6= t2 , respectively. In (3.39) diam(a, b, c) $ max(a, b, c) − min(a, b, c) is the diameter,
and the various constants are all positive.
Now, the discrete time counterpart of J 0 , the ∆LISDLG process ∆J(t), t ∈ Z, already exists and it is a
stationary process. As they are easy to obtain, the formulas corresponding to (3.38) and (3.39) in discrete
time are
^2 ∆21 |t|2H ,
^2 |t + 1|2H − 2 |t|2H + |t − 1|2H = const
cum(∆J(0), ∆J(t)) = const
2
for t 6= 0, where ∆ 12 f (x) = f x +
1
2
−f x −
1
2
is the central difference operator and ∆21 is its square, and
2
^3 |t + 1|2H − 2 |t|2H + |t − 1|2H cum(∆J(0), ∆J(t1 ), ∆J(t2 )) = const
t=diam(0,t1 ,t2 )
2H
2
^3 ∆ 1 |t| = const
,
2
(3.40)
t=diam(0,t1 ,t2 )
for t1 6= 0, t2 6= 0, t1 6= t2 , respectively.
Remark 13. It is interesting that in formula (3.40) the cumulant does not depend on the middle of the
points 0, t1 , t2 , it only depends on the diameter of the set of the three points. This property is stronger than
the following one.
cum(∆J(0), ∆J(t1 ), ∆J(t2 )) = cum(∆J(0), ∆J(t2 − t1 ), ∆J(t2 )),
(3.41)
for all t1 , t2 ∈ Z. The latter is equivalent to the fact that the bispectrum of the ∆LISDLG process ∆J(t) is
real-valued, see Remark 21.
Now we state two important properties of the finite dimensional distributions of the LISDLG process.
55
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
1
Theorem 8. The LISDLG process J(t) is c(p) = p 2(H−1) -dilative stable (see Def. 3), i.e. for every p > 0,
k ∈ N and t1 , . . . , tk > 0, the characteristic function ϕ of the k-dimensional distribution satisfies
p
ϕ(J(t1 ),... ,J(tk )) (u1 , . . . , uk ) = ϕ
−
p
1
2(H−1)
1
1
J p 2(H−1) t1 ,... ,J p 2(H−1) tk
(u1 , . . . , uk ),
i.e.
c2(H−1)
= ϕc−1 (J(ct1 ),... ,J(ctk )) (u1 , . . . , uk ),
ϕ(J(t1 ),... ,J(tk )) (u1 , . . . , uk )
(3.42)
for all u1 , . . . , uk ∈ R.
Proof. Let T ≥ max (t1 , . . . , tk ) and let us consider the function
. X
uj χ[tj ,T ] (s),
G(s) =
k
j=1
0 ≤ s ≤ T , where
k
X
|uj | < σ0−2 T −1 ,
j=1
and substitute it into (3.27). The replacement is legitimate, since
var(G) =
k
X
|uj | < σ0−2 T −1 ,
j=1
and for the cumulant function log ϕJ(t1 ),... ,J(tk ) (u1 , . . . , uk ) we have
log ϕJ(t1 ),... ,J(tk ) (u) = log ϕJ (G)
=
Z
∞
21−2H c0 X i` 2`
· 2
σ0
1 − H σ0
`
`=2
Dτ0 (s)2(H−1)
Ỳ
(G(T ) − G(sj )) ds
j=1
[0,T ]`
Z
Z
k
∞
21−2H c0 X i` 2` X
· 2
σ0
=
ui1 · · · ui` · · · Dτ0 (s)2(H−1) ds
1 − H σ0
`
i ,... ,i =1
ti1
`=2
for u∈ Rk :
∈R :
k
k
P
j=1
k
P
j=1
1
`
0
ti`
(3.43)
0
|uj | < σ0−2 T −1 . From (3.43) by a change of variable in the integral we have (3.42) for u
|uj | < σ0−2 T −1 .
Since the distribution of (J(t1 ), . . . , J(tk )) is infinitely divisible (see Cor. 3), the left hand side of (3.42)
is a characteristic function. On the other hand, the characteristic function on the right hand side of (3.42) is
uniquely determined by its values in any neighborhood of 0∈ Rk . This last fact follows easily if one considers
the proof of Theorem 6 (with JP = J) from formula (3.28). Thus we have proved (3.42) for all u∈ Rk .
Corollary 6. The finite-dimensional distributions of the LISDLG process J(t) are infinitely divisible.
Proof. It is a direct consequence of Theorem 8 or either of Corollary 3.
Hereafter several facts follow about the one-dimensional distributions of the increments of the LISDLG
process J(t). Because of the stationary increments, it suffices to examine the distribution of J(t) itself.
56
E. IGLÓI AND C. TERDIK
Proposition 15. The distribution of the random variable J(t) is uniquely determined by its cumulants
given by (3.33). For each t > 0 the logarithm of the characteristic function of J(t) has the series expansion

Z
∞
m
X
21−2H
 (iu)
2(m−1)
·
c
σ
log ϕJ(t) (u) =

0 0
m
1−H
m=2


Dτ0 (s)2(H−1) ds tm+2(H−1)  ,
[0,1]m
in the neighborhood u : |u| < σ0−2 t−1 of zero.
Proof. The cumulants and the series expansion have been calculated in the proof of Theorem 7. The
uniqueness has also been proved, see the proof of Theorem 6.
The next two propositions are special cases of Corollary 6 and of Theorem 8.
Proposition 16. The distribution of the random variable J(t) is infinitely divisible.
Proposition 17. For each t ∈ R+ the characteristic function of the random variable J(t)/t has the following
property, called one-dimensional dilative stability. (See Def. 3, Th. 8 and also Sect. 4.1 for an application.)
t2(H−1)
,
ϕ J(t) (u) = ϕJ(1) (u)
(3.44)
t
for all u ∈ R.
Actually the LISDLG process is derived from the SDLG process, which has Γ one-dimensional distribution.
So, one can expect that the one-dimensional distribution of the LISDLG process cannot be far from the shifted
Γ distribution. Now, we examine if it is so. If H → 1, then the m-fold integral in (3.33) converges to one.
Therefore if one sets the additional condition
c0
−→ c1 ,
2(1 − H) H→1
(3.45)
where c1 is a positive constant, then
2(m−1) m
cumm (J(t)) −→ (m − 1)!c1 σ0
H→1
t ,
which is the 2 6 m-th cumulant of the Γ(c1 /σ02 , σ02 t) distribution. Since the Γ distribution is uniquely
determined by its cumulants, we can state the following:
Remark 14. If H → 1 such that (3.45) holds, then the one-dimensional distribution of the shifted LISDLG
process converges to the Γ distribution, namely, we have
w
J(t) + c1 t −→ Γ(c1 /σ02 , tσ02 ).
H→1
(3.46)
Equation (3.46) means that the one-dimensional distribution of the ∆LISDLG process is easily mixed up
with a proper Γ distribution, leastwise for values of H near 1. See Section 4.2 for a potential occurrence in
practice.
3.3. Spectra of the SDLG process and its offsprings
The conventional definition of the spectrum of order m, see Definition 8, requires the integrability of
the m-th order joint cumulant as a function of m − 1 variables. This condition comes true in the case of
the stationary DLG process, e.g. see Lemma 2. However, LRD processes fail to satisfy it by definition, at
least for m = 2. If LRD is expressed also by the higher order cumulants, then the condition for the existence of
57
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
higher order spectra may be unfulfilled. This happens for the SDLG process Y (t), as the following proposition
states:
Proposition 18. For each m ∈ {2, 3, 4, . . . } the m-th order joint cumulant of the SDLG process Y (t), as a
function of m − 1 variables, is not integrable, i.e.
Z
|cum (Y (0), Y (t1 ), . . . , Y (tm−1 ))| dt(m−1) = ∞.
Rm−1
Proof. By formula (3.17),
Z
|cum (Y (0), Y (t1 ), . . . , Y (tm−1 ))| dt(m−1)
Rm−1
Z
2(H−1)
(2λ + |t1 | + |t2 − t1 | + · · · + |tm−1 − tm−2 | + |tm−1 |)
=K
Rm−1
dt(m−1)
Z
2(H−1)
> 22(H−1) K
(λ + |t1 | + |t2 | + · · · + |tm−1 |)
dt(m−1)
Rm−1
= ∞,
(3.47)
where K > 0 is a constant. The inequality in (3.47) is the consequence of the triangle inequality and that
2(H − 1) < 0, see Condition 1. The result of the last integral is infinity because of the Fubini theorem and
that H > 1/2, see Condition 1, too.
Fortunately, the integrability of a function guarantees the one-to-one correspondence between it and its
(inverse) Fourier transform. Thus Definition 8 can be generalized adequately for our case, i.e. when the
cumulant is not integrable.
Definition 9. Let m ∈ {2, 3, 4, . . . }. Let X(t), t ∈ R, be a stationary process with finite absolute moments
of order m, and let the cumulant function cum (X(0), X(t1 ), . . . , X(tm−1 )) be given. Assume that the
function Sm,X (ω (m−1) ) is integrable, and its inverse Fourier transform is just the cumulant function, i.e.
Z
cum (X(0), X(t1 ), . . . , X(tm−1 )) =
Rm−1

exp i
m−1
X

tj ωj  Sm,X (ω (m−1) )dω (m−1) .
j=1
Then function Sm,X (ω (m−1) ) is called the m-th order spectrum of process X(t). For m = 2 and m = 3 the
m-th order spectrum is called a spectrum and a bispectrum, respectively.
Thanks to Definition 9 we are now able to give the spectra in a clear form, leastwise the spectrum and
the bispectrum.
Proposition 19. The spectrum of the SDLG process Y (t) exists and it can be given either by the mixture
representation
S2,Y (ω) = EA S2,R1,1 (ω),
(3.48)
or by the following explicit expression.


Z∞
1
1
·
µ0 λ3−2H σ02 Re eiλω x2(H−1) e−ixω dx .
S2,Y (ω) =
2π 1 − H
λ
(3.49)
58
E. IGLÓI AND C. TERDIK
Proof. Because of the mixture representation of the cumulant function, see Lemma 4, we have
cum (Y (0), Y (t)) = EA cum (R1,1 (0), R1,1 (t))
Z
= EA eitω S2,R1,1 (ω)dω
R
Z
eitω EA S2,R1,1 (ω)dω
=
R
=
1
2π
1
=
2π
=
1
2π
Z
eitω EA
R
Z
e
2 3−2H
itω 2µ0 σ0 λ
dω
Z∞
1
x2(1−H) e−λx dxdω
ω 2 + x2
0


Z∞
µ0 λ3−2H σ02 Re eiλω e−isω s2(H−1) ds dω.
Γ(3 − 2H)
R
Z
eitω
R
2µ0 σ02
2
ω + 4α2
1
1−H
(3.50)
λ
The third equation is true because of the Fubini theorem, the fourth is the consequence of Proposition 8,
formula (2.3), Condition 4 and of the fact that µ = µ0 . The fifth equation is motivated by Condition 3.
Define the function


Z∞
1
1
·
µ0 λ3−2H σ02 Re eiλω e−isω s2(H−1) ds
S2,Y (ω) $
2π 1 − H
=
λ3−2H
1
2µ0 σ02
2π
Γ(3 − 2H)
Z∞
λ
1
x2(1−H) e−λx dx.
ω 2 + x2
(3.51)
0
By the second expression in (3.51), changing the order of the integrals we have
Z
R
1
λ3−2H
2µ0 σ02
|S2,Y (ω)| dω =
2π
Γ(3 − 2H)
= µ0 σ02
3−2H
λ
Γ(3 − 2H)
Z∞
Z∞ Z
0
R
ω2
1
dωx2(1−H) e−λx dx
+ x2
x1−2H e−λx dx = µ0 σ02 λ
0
1
< ∞.
2(1 − H)
Therefore S2,Y (ω) is integrable and its inverse Fourier transform is the second order cumulant function,
see formula (3.50). Thus, S2,Y (ω) is really the spectrum of process Y (t), and this proves formula (3.49).
Formula (3.48) is the consequence of the third equation in (3.50) and the one-to-one correspondence between
the spectrum and the covariance.
Proposition 20. Recall that the LISDLG process J(t) has stationary increments. Thus if one defines the
d
J as the limit
spectrum of the informal derivative process J 0 = dt
S2,J 0 (ω) $ lim S2,Yλ (ω),
λ→0
(3.52)
then one has
S2,J 0 (ω) =
1
1 Γ(2H − 1)
1−2H
1−2H
c0 σ02 |ω|
·
c0 σ02 sin(Hπ) |ω|
=
.
2 cos(Hπ)Γ(3 − 2H)
2π
1−H
(3.53)
We will call S2,J 0 (ω) the generalized spectrum (appertaining to process J(t) [to avoid reference to the informal
derivative process J 0 ]).
59
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
Proof. Applying Proposition 19 and Condition 7 we have
∞

Z
1
1
·
c0 σ02 Re  e−isω s2(H−1) ds
S2,J 0 (ω) = lim S2,Yλ (ω) =
λ→0
2π 1 − H
0
1
1 Γ(2H − 1)
1
1−2H
·
c0 σ02 Re Γ(2H − 1)(iω)1−2H =
·
c0 σ02 sin(Hπ) |ω|
,
2π 1 − H
2π
1−H
=
where the third equation is the consequence of formula (2.3.3.1) in [30]. By the identity
Γ(2H − 1)Γ(2(1 − H)) = −
π
,
sin(2πH)
(3.54)
one obtains the constant multiplier in the required form.
Remark 15. In view of Corollary 5, it is not surprising that, apart from a constant multiplier, S2,J 0 (ω) in
Corollary 20 is the same as the informal spectrum of the continuous time FGN, i.e. the informal spectrum
of the informal derivative of the FBM.
Let us consider the analogous statements for the bispectrum.
Proposition 21. The bispectrum of the SDLG process Y (t) exists and it can be given either by the mixture
representation
S3,Y ω(2) = EA S3,R1,1 ω (2) ,
(3.55)
or in the following explicit manner.
S3,Y ω(2) = −
1
2π
2

3−2H
6µ0 λ
1−H
σ04

Z∞
1
Im sym ω12 ei(λ−s)ω1 s2(H−1) ds ,
ω1 ω2 ω3
(ω(3) )
λ
(3.56)
where ω3 $ −ω1 − ω2 . The bispectrum S3,Y ω(2) has the extraordinary property that it is real-valued moreover it is positive.
Proof. Because of the mixture representation of the cumulant function, see Lemma 4, we have
cum (Y (0), Y (t1 ), Y (t2 )) = EA cum (R1,1 (0), R1,1 (t1 ), R1,1 (t2 ))
Z
Z
= EA ei(t1 ω1 +t2 ω2 ) S3,R1,1 (ω1 , ω2 )dω (2) = ei(t1 ω1 +t2 ω2 ) EA S3,R1,1 ω(2) dω (2)
=
R2
1
2π
2 Z
R2

R2
1
sym
ei(t1 ω1 +t2 ω2 ) 12µ0 σ04 EA 
−2α (ω )
(3)

1
1
 dω(2) .
·
−iω1 − 2α iω2 − 2α
The third equation is due to the Fubini theorem, and the fourth is the consequence of Proposition 9,
formula (2.28), Condition 4, and of the fact that µ = µ0 . Let us define the function
S3,Y ω (2) $ EA S3,R1,1 ω (2)
2
1
1
1
1
·
·
12µ0 σ04 sym EA
·
=
2π
−2α −iω1 − 2α iω2 − 2α
(ω(3) )
(3.57)
60
E. IGLÓI AND C. TERDIK
Notice that S3,Y ω (2) must be real-valued and positive, because for each realization of the random variable
−2α, so is the bispectrum S3,R1,1 ω(2) . Now we have to calculate the expectation
E
1
1
1
·
·
−2α −iω1 − 2α iω2 − 2α
,
where −2α ∼ Γ(3 − 2H, 1/λ), see Condition 3. Function 1/(iω − 2α), ω ∈ R, is just the Fourier transform of
function exp(2αt), t > 0, and the latter is more usable from the point of view of the Γ distribution. Using
this observation, and partial fraction decomposition, one can easily obtain
E
1
1
1
·
·
−2α −iω1 − 2α iω2 − 2α
3−2H
=λ
1
ω1 ω2
Z∞ 1−
λ
ω1
ω2
e−i(λ−s)ω1 −
ei(λ−s)ω2
ω1 + ω2
ω1 + ω2
s2H−3 ds.
Substituting this expression into (3.57), we have
S3,Y ω(2)
=
=
=
=
1
2π
1
2π
1
2π
1
2π
2

1
12µ0 λ3−2H σ04 sym 
ω
1 ω2
(ω )
(3)
2
3−2H
12µ0 λ
σ04
2
12µ0 λ3−2H σ04
2
12µ0 λ3−2H σ04
1
ω1 ω2 ω3
1
ω1 ω2 ω3
1
ω1 ω2 ω3
Z∞ 1−
λ
ω1
ω2
e−i(λ−s)ω1 −
ei(λ−s)ω2
ω1 + ω2
ω1 + ω2

s2H−3 ds
λ
ω2 ω3 −i(λ−s)ω1
ω1 ω3 i(λ−s)ω2 2H−3
sym −
e
−
e
ds
s
ω1 + ω2
ω1 + ω2
(ω(3) )
λ
sym ω2 ei(λ−s)ω1 + ω1 ei(λ−s)ω2 s2H−3 ds
(ω(3) )
λ
sym −ω1 ei(λ−s)ω1 s2H−3 ds.
(ω(3) )
Z∞
Z∞
Z∞
(3.58)
Integrating by parts changes (3.58) into
S3,Y ω(2) = −
1
2π
2


Z∞
6µ0 λ3−2H 4
1
σ0
Im sym ω12 ei(λ−s)ω1 s2(H−1) ds .
1−H
ω1 ω2 ω3
(ω(3) )
λ
What is left to prove is that S3,Y ω (2) is integrable.
Z
Z
Z S3,Y ω(2) dω (2) = S3,Y ω(2) dω (2) = EA S3,R1,1 ω(2) dω (2)
R2
R2
Z
= EA
R2
S3,R1,1 ω (2) dω(2) = EA cum3 (R1,1 (0)) = 2µ0 σ04 EA
R2
= µ0 σ04
1
−2α
λ
< ∞.
1−H
Here the first equation is the consequence of the positivity of S3,Y ω(2) , the second is true because of (3.57),
the third is the Fubini theorem, the fourth is the consequence of Definition 8, the fifth is true because of
Proposition 6, Condition 4 and of the fact that µ = µ0 . The sixth equation is motivated by Condition 3,
and the finiteness arises from inequality H < 1, see Condition 1.
61
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
Proposition 22. If one defines the bispectrum of the informal derivative process J 0 =
d
dt J
as the limit
S3,J 0 ω (2) $ lim S3,Yλ ω (2) ,
(3.59)
λ→0
then one obtains
1 2 6Γ(2H − 1)
1
c0 σ04
Im sym (iω1 )3−2H ,
S3,J 0 ω (2) =
2π
1−H
ω1 ω2 ω3
(ω(3) )
(3.60)
or, in another form,
3
|ω1 |2(1−H)
1
·
c0 σ04 sym
,
S3,J 0 ω (2) =
2π sin(Hπ)Γ(3 − 2H)
(ω ) ω2 ω3
(3.61)
(3)
where ω3 $ −ω1 − ω2 .
We will call S3,J 0 (ω (2) ) the generalized bispectrum (appertaining to process J(t) [to avoid reference to the
informal derivative process J 0 ]).
The generalized bispectrum S3,J 0 ω (2) has the extraordinary property that it is real-valued moreover it
is positive.
Proof. The bispectrum of the SDLG process Y (t) is
S3,Y ω(2) = −
1
2π
2


Z∞
6µ0 λ3−2H 4
1
σ0
Im sym ω12 ei(λ−s)ω1 s2(H−1) ds ,
1−H
ω1 ω2 ω3
(ω(3) )
λ
(3.62)
see (3.56). Though in (3.62) the function to be integrated is not Lebesgue integrable on the whole positive
ray, the integral exists in the sense of principal value. In particular, the following integral is defined, and
there is an explicit expression for it, i.e.
Z∞
ZN
e−isω s2(H−1) ds $ lim
N →∞
0
e−isω s2(H−1) ds = Γ(2H − 1)(iω)1−2H ,
(3.63)
0
ω 6= 0, see formula (2.3.3.1) in [30]. By (3.59) we have to take the limit as λ → 0 in (3.62). To do this, cut
the integral of (3.62) into two parts:
Z∞
Z1
e
i(λ−s)ω 2(H−1)
s
ds =
λ
Z∞
e
i(λ−s)ω 2(H−1)
s
ds + e
iλω
e−isω s2(H−1) ds.
(3.64)
1
λ
By Condition 7 and (3.64) the convergence of (3.62) to (3.59) is now obvious. In another form we have
2(1−H)
|ω1 |
3
1
·
c0 σ04 sym
,
S3,J 0 ω(2) =
2π sin(Hπ)Γ(3 − 2H)
( ω ) ω2 ω3
(3)
where we also applied identity
(3.54).
The fact that S3,J 0 ω (2) is real-valued and positive, follows from its form (3.61) since
2(1−H)
sym
(ω(3) )
|ω1 |
ω2 ω3
3−2H
=
|ω1 |
3−2H
sig(ω1 ) + |ω2 |
3−2H
sig(ω2 ) + |ω3 |
3ω1 ω2 ω3
sig(ω3 )
> 0,
(3.65)
62
E. IGLÓI AND C. TERDIK
for all ω1, ω2 ∈ R for which ω1 ω2 ω3 6= 0. The inequality in (3.65) is easily obtainable supposing without loss
of generality that ω1 ω2 > 0 and using the inequality (x + 1)3−2H > 1 + x3−2H with x = |ω2 /ω1 |.
Remark 16. The generalized bispectrum S3,J 0 (ω (2) ) behaves around zero as
S3,J 0 (ω1 , ω2 )
'
ω1 ,ω2 →0
q
−2H
ω12 + ω22
.
Remark 17. The property of the bispectrum S3,Y ω(2) that it is real-valued, is equivalent to the property
of the cumulants that
cum (Y (0), Y (t1 ), Y (t2 )) = cum (Y (0), Y (t2 − t1 ), Y (t2 )) ,
(3.66)
t1 , t2 ∈ R.
Proof. By the stationarity we have
cum (Y (0), Y (−t1 ), Y (−t2 )) = cum (Y (t2 ), Y (t2 − t1 ), Y (0)) = cum (Y (0), Y (t2 − t1 ), Y (t2 )) .
(3.67)
If the bispectrum is real, then by Definition 9 we have
cum (Y (0), Y (t1 ), Y (t2 )) = cum (Y (0), Y (−t1 ), Y (−t2 )) .
Comparing (3.68) and (3.67) we have (3.66).
On the other hand, if (3.66) holds, then by Definition 9 we have
Z
Z
ei(t1 ω1 +t2 ω2 ) S3,Y ω (2) dω (2) = ei(t1 ω1 +t2 ω2 ) S3,Y ω (2) dω (2) ,
R2
(3.68)
(3.69)
R2
for all t1 , t2 ∈ R. There is a one-to-one correspondence between integrable functions and their inverse Fourier
transforms, thus by (3.69) the bispectrum must be real.
Incidentally, equation (3.66) is also a consequence of the form of the cumulants, see Lemma 4. More
generally, the following statement holds true.
Remark 18. We have
cum (Y (0), Y (t1 ), . . . , Y (tm−1 )) = cum (Y (0), Y (tm−1 − tm−2 ), . . . , Y (tm−1 )) ,
for all m ≥ 3 and t1 , . . . , tm−1 ∈ R.
Proof. The terms in the sum in (3.17) correspond to circles (Y (0), Y (ti1 ), . . . ,Y (tim−1 ), Y (0)), and any such
term remains the same if one walks round the circle in the opposite direction.
In practice it is usually not the whole process but its discretely sampled version that is observed. For the
LISDLG process J(t) this means that we deal with the discrete time LISDLG process J(t), t ∈ Z, say, or
equivalently, with the ∆LISDLG process ∆J(t) = J(t + 1) − J(t), t ∈ Z. The latter is more desirable for the
very reason that it is stationary. Unfortunately, its continuous time counterpart, the derivative process J 0 (t)
does not exist, and thus there is another reason why we have to deal with the ∆LISDLG process ∆J(t),
t ∈ Z. On the other hand, we have seen above that the bispectrum of the SDLG process Yλ (t) is real-valued
and positive. We will state that the discrete time counterpart of the SDLG process Yλ (t), t ∈ R, i.e. the
∆ISDLG process ∆Jλ (t), t ∈ Z, has a bispectrum that is also real-valued and positive. Going further and
taking the limit as λ → 0, we arrive at the ∆LISDLG process ∆J(t), t ∈ Z, and we find that it also has a
bispectrum which is real-valued and positive too.
It is important to remark that in the case of discrete time LRD processes we will use the notion of
spectrum of any order in the same sense as in the case of continuous time LRD processes. That is, even if the
cumulant series is not absolutely summable, the spectrum must be integrable and its inverse Fourier transform
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
63
must be the cumulant series. Clearly, the integrability of the spectrum ensures – also in the discrete-time
case – a one-to-one correspondence between the spectrum and the cumulant series.
Let us examine first the second order spectrum.
Proposition 23. The spectrum of the ∆ISDLG process ∆Jλ (t) exists and it is
∞ −2
2 X
S2,Yλ ω (k)
ω (k)
S2,∆Jλ (ω) = eiω − 1
k=−∞



Z∞
∞
−2
X
2
(k)
(k)
1 µ0 λ3−2H 2 iω
 ω (k)
·
σ e − 1 Re 
eiλω
e−ixω x2(H−1) dx ,
=
2π
1−H 0
k=−∞
λ
for ω ∈ (0, 2π), where ω (k) $ ω + 2kπ, k ∈ Z.
Proof. Let us calculate cum (∆Jλ (0), ∆Jλ (0)). At first, we can change the order of the cumulant operator
and the integrals by right of a similar reasoning as in (2.26) (see the explanation after (2.26)). Then applying
Lemma 4 we have

cum (∆Jλ (0), ∆Jλ (0)) = cum 
Z1

Z1
Yλ (s)ds
Yλ (s)ds,
0
0
Z1 Z1
cum (Yλ (s0 ), Yλ (s1 )) ds1 ds0
=
0
0
Z1 Z1
σ02
=
0
0
µ0 λ
2(1 − H)
2(H−1)
Dτ (s)
ds1 ds0
1+
2λ
1
σ 2 µ0 λ3−2H
=
2(1 − H) 0
Z1 Z1
2(H−1)
(λ + |s1 − s0 |)
0
ds1 ds0
0
1 1
2H
2
3−2H
2H
2H−1
σ µ0 λ
(λ + 1) −λ
=
−λ
.
(1 − H) (2H − 1) 0
2H
(3.70)
We will need the resulting expression later, but presently we only use the fact that cum (∆Jλ (0), ∆Jλ (0)) is
finite.
Now let us calculate cum (∆Jλ (0), ∆Jλ (t)) in general. We can change the order of the cumulant operator
and the integrals as above, so by the stationarity we have

cum (∆Jλ (0), ∆Jλ (t)) = cum 
Z1
0

Zt+1
Yλ (s)ds,
Yλ (s)ds
t
Z1 Zt+1
cum (Yλ (0), Yλ (s1 − s0 )) ds1 ds0 .
=
0 t
(s0 )(s1 )
(3.71)
64
E. IGLÓI AND C. TERDIK
We know that S2,Yλ (ω) is the spectrum of the continuous time process Yλ (t), therefore continuing (3.71), we
obtain
Z1 Zt+1Z
cum (∆Jλ (0), ∆Jλ (t)) =
ei(s1 −s0 )ω S2,Yλ (ω)dωds1 ds0
0 t R
(s0 )(s1 )(ω)
Z Z1 Zt+1
=
ei(s1 −s0 )ω ds1 ds0 S2,Yλ (ω)dω,
(3.72)
R 0 t
(ω)(s0 )(s1 )
where the change of the order of the integrals is the consequence of the integrability of S2,Yλ (ω). Calculating
the two inner integrals in the last row of (3.72), and breaking the outer integral and performing the proper
substitutions yields
Z
cum (∆Jλ (0), ∆Jλ (t)) =
2
eitω ω −2 eiω − 1 S2,Yλ (ω)dω
R
∞ Z2π
X
=
−2 eiω − 12 S2,Y ω (k) dω.
eitω ω (k)
λ
(3.73)
k=−∞ 0
For t = 0 the above double integral (i.e. a sum and an integral) is finite, because cum (∆Jλ (0), ∆Jλ (0)) is
finite. Thus by the Fubuni theorem we can change the order of the sum and the integral in (3.73) to get
Z2π
cum (∆Jλ (0), ∆Jλ (t)) =
e
itω
∞ −2
iω
X
(k)
(k)
e − 12
ω
S2,Yλ ω
dω.
(3.74)
k=−∞
0
Let us define the function
∞ −2
iω
2 X
(k)
(k)
ω
S2,Yλ ω
,
S2,∆Jλ (ω) $ e − 1
(3.75)
k=−∞
ω ∈ (0, 2π). We have just mentioned the integrability of S2,∆Jλ (ω), thus the spectrum of the discrete time
process ∆Jλ (t) really exists and by (3.74) it is the above-defined S2,∆Jλ (ω). Moreover, by substituting the
form (3.49) of S2,Yλ (ω) into (3.75), we have



Z∞
∞
−2
2 X
(k)
(k)
1 µ0 λ3−2H 2 iω
 ω (k)
·
σ e − 1
Re eiλω
e−ixω x2(H−1) dx .
S2,∆Jλ (ω) =
2π
1−H 0
k=−∞
λ
The following technical concept will be needed in some of subsequent proofs:
Definition 10. We will say that a set of functions fλ (ω) converges as λ → 0 essentially monotonely to a
function f (ω), if there exist a set of constants cλ , a constant c, a set of functions gλ (ω) and a function g(ω)
such that fλ (ω) = cλ gλ (ω) for all λ and ω, f (ω) = cg(ω) for all ω, moreover cλ −→ c, and the convergence
fλ (ω) −→ f (ω) is monotone (either increasing for all ω or decreasing for all ω).
λ→0
λ→0
65
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
Theorem 9. The spectrum of the ∆LISDLG process ∆J(t) exists and it is
∞ −2
iω
2 X
(k)
(k)
0
ω
S2,J ω
S2,∆J (ω) = e − 1
k=−∞
∞ −1−2H
X
iω (k)
2
1 −Γ(2H − 1)
·
c0 σ02 eiω − 1 Re
=
2π
1−H
=
k=−∞
∞
2 X
1 Γ(2H − 1)
·
sin(Hπ)c0 σ02 eiω − 1
2π
1−H
!
(k) −1−2H
,
ω (3.76)
k=−∞
for ω ∈ (0, 2π), where ω (k) $ ω + 2kπ, k ∈ Z.
Proof. By (3.51),
2
1
·
µ0 λ3−2H σ02
S2,Yλ (ω) =
2π Γ(3 − 2H)
Z∞
x2(1−H) e−λx
1
dx,
ω 2 + x2
0
thus by (3.52) and Condition 7
2
1
·
c0 σ02
S2,J 0 (ω) = lim S2,Yλ (ω) =
λ→0
2π Γ(3 − 2H)
Z∞
x2(1−H)
1
dx.
ω 2 + x2
(3.77)
0
Moreover, the convergence
Z∞
2(1−H) −λx
x
e
1
dx −→
λ→0
ω 2 + x2
0
Z∞
x2(1−H)
1
dx
ω 2 + x2
(3.78)
0
is monotone, that is, for each ω 6= 0, the left-hand side of (3.78) increases as λ decreases. So thus, the
convergence in (3.77) is essentially monotone.
Now, by Proposition 23 and formula (3.53),
∞ −2
X
2
S2,Yλ ω (k)
ω (k)
lim S2,∆Jλ (ω) = eiω − 1 lim
λ→0
λ→0
2
= eiω − 1
k=−∞
∞
X ω
(k)
−2
S2,J 0 ω
(k)
k=−∞
∞ 2 X
1 Γ(2H − 1)
(k) −1−2H
2 iω
·
sin(Hπ)c
e
−1
,
=
ω 0 σ0
2π
1−H
(3.79)
k=−∞
which is finite for all ω ∈ (0, 2π). In (3.79) the change of order of the limit and the infinite sum can be justified
by the Beppo Levi theorem and the fact that the convergence (3.77) is essentially monotone. Therefore let
us define the function
S2,∆J (ω) $ lim S2,∆Jλ (ω),
λ→0
(3.80)
66
E. IGLÓI AND C. TERDIK
ω ∈ (0, 2π). Then S2,∆J (ω) is integrable, because by the Fatou lemma we have
Z2π
Z2π
|S2,∆J (ω)| dω =
lim |S2,∆Jλ (ω)| dω
0
0
λ→0
Z2π
6 lim
λ→0
S2,∆Jλ (ω)dω
0
= lim cum (∆Jλ (0), ∆Jλ (0))
λ→0
1
1 2H
2
3−2H
2H
2H−1
σ0 µ0 λ
(λ + 1) − λ
−λ
= lim
2H
λ→0 (1 − H) (2H − 1)
1
σ 2 c0 < ∞.
(3.81)
=
(1 − H) (2H − 1)2H 0
The fourth row in (3.81) is the consequence of formula (3.70). Moreover,
Z2π
Z2π
e
0
itω
eitω lim S2,∆Jλ (ω)dω
S2,∆J (ω)dω =
λ→0
0
Z2π
eitω S2,∆Jλ (ω)dω = lim cum (∆Jλ (0), ∆Jλ (t)) = cum (∆J(0), ∆J(t)) ,
= lim
λ→0
λ→0
(3.82)
0
where the second equation can be justified by applying the Beppo Levi theorem both to the real and to the
imaginary parts and considering that the convergence in (3.80) is essentially monotone. The last equation
in (3.82) is the consequence of Theorem 6. Thus, S2,∆J (ω) is really the spectrum of the discrete time process
∆J(t). The first equation of (3.76) is the second equation of (3.79). The second equation of (3.76) arises
from Proposition 23, since
S2,∆J (ω) = lim S2,∆Jλ (ω)
λ→0



Z∞
∞
X
−2
(k)
1
1
2
 ω (k)
·
c0 σ02 eiω − 1 Re 
e−ixω x2(H−1) dx
=
2π 1 − H
k=−∞
0
!
∞ −1−2H X
1 −Γ(2H − 1)
2
2 iω
(k)
·
c0 σ0 e − 1 Re
=
iω
,
2π
1−H
k=−∞
where we also used formula (2.3.3.1) in [30]. At last, the third equation of (3.76) is just the last row of
formula (3.79).
Remark 19. The spectrum of the ∆LISDLG process ∆J(t) is the same as that of the FGN, apart from
some multiplicative constant.
Let us turn now to the bispectrum.
67
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
Proposition 24. The bispectrum of the ∆ISDLG process ∆Jλ (t) exists and it is
∞ −1
X
(k) (`) (k+`)
(k,`)
ω1 ω2 ω3
S3,Yλ ω (2)
3
∞
Y
X
(1 − eiωj )
S3,∆Jλ ω (2) = −i
=
× Im

∞
 X

j=1
k=−∞ `=−∞
2
3
Y
6iµ0 λ
σ04
(1 − eiωj )
1−H
j=1



Z∞
∞

2
X (k) (`) (k+`) −2
(k)
(k)
 ω1 ω2 ω3
 ω1(k) eiλω1
e−ixω1 x2(H−1) dx ,
sym

(k,`)
1
2π
3−2H
ω (3)
k=−∞ `=−∞
λ
(k)
∈ (0, 2π)2 , such that ω3 6= 0, where ω3 $ −ω1 − ω2 , ωj $ ωj + 2kπ, j = 1, 2, k ∈ Z, and
(k+`)
(k)
(`)
(k,`)
(k)
(`)
(k,`)
(k)
(`)
(k+`)
$ −ω1 − ω2 , ω(2) $ ω1 , ω2 , ω(3) $ ω1 , ω2 , ω3
, k, ` ∈ Z.
ω3
The bispectrum S3,∆Jλ ω(2) has the extraordinary property that it is real-valued and positive.
for
ω(2)
Proof. Not only do the statements of this proposition and Proposition 23 correspond but the proofs are also
very similar.
Let us calculate cum (∆Jλ (0), ∆Jλ (0), ∆Jλ (0)) . At first, we can change the order of the cumulant operator
and the integrals by right of a similar reasoning as in (2.26) (see the explanation after (2.26)). Then applying
Lemma 4 yields

cum (∆Jλ (0), ∆Jλ (0), ∆Jλ (0)) = cum 
Z1
Z1
Yλ (s)ds,
0
Z1
Yλ (s)ds,
0

Yλ (s)ds
0
Z1 Z1 Z1
cum (Yλ (s0 ), Yλ (s1 ), Yλ (s2 )) ds2 ds1 ds0
=
0
0
0
Z1 Z1 Z1
σ04
=2
0
0
0
µ0 λ
2(1 − H)
1
σ 4 µ0 λ3−2H 3!
=
1−H 0
2(H−1)
Dτ0 (s)
ds
1+
2λ
Z1 Z1 Z1
(λ + s2 − s0 )
0 s0 s1
(s0 )(s1 )(s2 )
=
3λ3−2H µ0 σ04
(1 − H)(2H − 1)H
2(H−1)
ds
2 2H+1
λ
−(λ+1)2H+1 +(λ+1)2H +λ2H
2H + 1
.
(3.83)
We will need the resulting expression later, but presently we use only the fact that cum (∆Jλ (0), ∆Jλ (0) ,
∆Jλ (0)) is finite.
68
E. IGLÓI AND C. TERDIK
Now let us calculate cum (∆Jλ (0), ∆Jλ (t1 ), ∆Jλ (t2 )) in general. We can change the order of the cumulant
operator and the integrals as above, so by the stationarity, we have, that

cum (∆Jλ (0), ∆Jλ (t1 ), ∆Jλ (t2 )) = cum 
Z1
tZ
1 +1
Yλ (s)ds,

Yλ (s)ds
Yλ (s)ds,
t1
0
tZ
2 +1
t2
Z1 tZ1 +1tZ2 +1
cum (Yλ (0), Yλ (s1 − s0 ), Yλ (s2 − s0 )) ds.
=
(3.84)
t2
0 t1
(s0 ) (s1 ) (s2 )
We know that S3,Yλ ω (2) is the bispectrum of the continuous time process Yλ (t), hence by continuing
(3.84) we have
Z1 tZ1 +1tZ2 +1 Z
cum (∆Jλ (0), ∆Jλ (t1 ), ∆Jλ (t2 )) =
ei((s1 −s0 )ω1 +(s2 −s0 )ω2 ) S3,Yλ ω (2) dω (2) ds
t2
0 t1
R2
(s0 ) (s1 ) (s2 ) (ω )
(2)
Z
Z1 tZ1 +1tZ2 +1
ei((s1 −s0 )ω1 +(s2 −s0 )ω2 ) dsS3,Yλ ω (2) dω (2) ,
=
R2
(ω(2)
(3.85)
t2
0 t1
)(s0 ) (s1 ) (s2 )
where the change of order of the integrals is the consequence of the integrability of S3,Yλ ω(2) . Calculating
the three inner integrals in the last row of (3.85), then cutting up the outer integral and performing the
proper substitutions, we arrive at
Z
cum (∆Jλ (0), ∆Jλ (t1 ), ∆Jλ (t2 )) = −i
=
∞
X
∞
X
Z
ei(t1 ω1 +t2 ω2 ) (−i)
ei(t1 ω1 +t2 ω2 )
R2
3
Y
3
Y
(1 − eiωj )ω1−1 ω2−1 ω3−1 S3,Yλ ω(2) dω (2)
j=1
−1
(k) (`) (k+`)
(k,`)
(1 − eiωj ) ω1 ω2 ω3
S3,Yλ ω(2) dω (2) .
(3.86)
j=1
k=−∞ `=−∞
[0,2π]2
Remembering that S3,Yλ ω (2) is real-valued and positive, one can easily check that the function
3
−1
Y
(k) (`) (k+`)
(k,`)
(1 − eiωj ) ω1 ω2 ω3
S3,Yλ ω (2)
f k, l, ω(2) = −i
(3.87)
j=1
is real and positive, too. Thus
Z ∞
∞
3
−1
X
X
Y
i(t ω +t ω )
(k) (`) (k+`)
(k,`)
e 1 1 2 2 (−i)
(1 − eiωj ) ω1 ω2 ω3
S3,Yλ ω (2) dω (2)
j=1
k=−∞ `=−∞
2
[0,2π]
=
∞
X
∞
X
Z
f k, l, ω(2) dω (2) = cum (∆Jλ (0), ∆Jλ (0), ∆Jλ (0)) ,
(3.88)
k=−∞ `=−∞
[0,2π]2
where the last equation is (3.86) for t1 = t2 = 0. The right-hand side of (3.88) is finite, see the beginning
of this proof. Therefore by the Fubini theorem we can change the order of the double sum and the double
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
69
integral in (3.86) to get
cum (∆Jλ (0), ∆Jλ (t1 ), ∆Jλ (t2 ))
Z
3
∞
Y
X
ei(t1 ω1 +t2 ω2 ) (−i)
(1 − eiωj )
=
[0,2π]
j=1
2
∞ X
(k)
(`)
(k+`)
ω1 ω2 ω3
−1
(k,`)
S3,Yλ ω(2)
dω (2) .
(3.89)
(k,`)
S3,Yλ ω (2)
,
(3.90)
k=−∞ `=−∞
Let us define the function
3
∞
Y
X
(1 − eiωj )
S3,∆Jλ ω (2) $ −i
j=1
∞ X
(k)
(`) (k+`)
ω1 ω2 ω3
−1
k=−∞ `=−∞
for ω(2) ∈ (0, 2π)2 such that ω3 6= 0. We have just mentioned the integrability of S3,∆Jλ ω(2) , thus, the
bispectrum of the discrete time process ∆Jλ (t) really exists and by (3.89) it is S3,∆Jλ ω(2) . Moreover, by
substituting the form (3.56) of S3,Yλ ω(2) into (3.90), we have
S3,∆Jλ ω(2) =
1
2π
2
3−2H
6iµ0 λ
1−H
σ04
3
Y
j=1
(1 − eiωj ) Im

∞
 X

∞
X
k=−∞ `=−∞



Z∞

−2
2
(k)
(k)
(k)
(`)
(k+`)
(k)
iλω1
−ixω1
2(H−1)
 ·
 ω1 ω2 ω3
 ω1
sym
e
e
x
dx
(k,`) 
ω (3)
λ
The fact that S3,∆Jλ ω (2) is real-valued and positive, follows from its form (3.90) since function f k, l, ω(2)
has the same properties.
Theorem 10. The bispectrum of the ∆LISDLG process ∆J(t) exists and it is
S3,∆J ω (2) = −i
3
Y
(1 − e
iωj
)
j=1
∞
X
∞ −1
X
(k) (`) (k+`)
(k,`)
ω1 ω2 ω3
S3,J 0 ω (2)
k=−∞ `=−∞
×
3
Y
j=1
(1 − eiωj ) Im
2
−6iΓ(2H − 1)
1
c0 σ04
2π
1−H


∞
−2
3−2H 
X
(k)
 ω1(k) ω2(`) ω3(k+`)

iω1
sym

(k,`)
=

∞
 X

k=−∞ `=−∞
ω (3)


(k) 2(1−H)
∞
3
∞
ω
Y
X
X
−1
1
−3i
 (k) (`) (k+`)

c0 σ04
(1 − eiωj )
=
 ω1 ω2 ω3
sym
(`) (k+`)  ,
2π sin(Hπ)Γ(3 − 2H)
(k,`)
ω2 ω3
ω
j=1
k=−∞ `=−∞
(3.91)
(3)
(k)
∈ (0, 2π)2 such that ω3 6= 0, where ω3 $ −ω1 − ω2 , ωj $ ωj + 2kπ, j = 1, 2, k ∈ Z, and
(k+`)
(k)
(`)
(k,`)
(k)
(`)
(k,`)
(k)
(`)
(k+`)
$ −ω1 − ω2 , ω (2) $ ω1 , ω2 , ω(3) $ ω1 , ω2 , ω3
, k, ` ∈ Z.
ω3
The bispectrum S3,∆J ω (2) has the extraordinary property that it is real-valued moreover it is positive.
for
ω (2)
70
E. IGLÓI AND C. TERDIK
Proof. By (3.57) and Condition 3,
S3,Yλ ω(2) = EA S3,R1,1 ω (2)
=
1
2π
2
Z∞
12
µ0 λ3−2H σ04
Γ(3 − 2H)
2(1−H) −λx
x
e
0
sym
(ω(3) )
1
1
1
·
·
x −iω1 + x iω2 + x
dx,
hence by (3.59) and Condition 7,
S3,J 0 ω(2) = lim S3,Yλ ω(2)
λ→0
=
1
2π
2
12
c0 σ04
Γ(3 − 2H)
Z∞
x
sym
(ω(3) )
1
1
1
·
·
x −iω1 + x iω2 + x
Z∞
2(1−H)
0
dx.
(3.92)
Moreover, the convergence
Z∞
0
x2(1−H) e−λx sym
(ω(3) )
1
1
1
·
·
x −iω1 + x iω2 + x
dx −→
λ→0
0
x2(1−H) sym
(ω(3) )
1
1
1
·
·
x −iω1 + x iω2 + x
dx
(3.93)
is monotone, that is, for every ω1 , ω2 ∈ R such that ω1 ω
2 ω3 6=0, the left-hand side of (3.93) increases as λ
decreases. The motivation is the nonnegativity of S3,R1,1 ω(2) and thus of the symmetrized term in (3.93).
Observe also that the convergence in (3.92) is essentially monotone.
At this point we anticipate that for each (k, `) ∈ Z2 the term −i
3
Q
−1
(k) (`) (k+`)
(1 − eiωj ) ω1 ω2 ω3
is
j=1
real-valued and positive, as it is easy to see.
By Proposition 24 and formulae (3.60) and (3.61), we have
∞ X
∞
3
Y
X
(1 − eiωj ) lim
lim S3,∆Jλ ω (2) = −i
λ→0
λ→0
j=1
= −i
3
Y
(1 − eiωj )
j=1
(k)
(`)
(k+`)
ω1 ω2 ω3
−1
(k,`)
S3,Yλ ω(2)
k=−∞ `=−∞
∞ −1
X
(k) (`) (k+`)
(k,`)
ω1 ω2 ω3
S3,J 0 ω(2)
∞
X
k=−∞ `=−∞
2
−6ic0 σ04 Γ(2H − 1)
1
=
2π
1−H



∞
3
∞
 X
−2
3−2H 
Y
X
(k)
 ω1(k) ω2(`) ω3(k+`)

iω1
(1 − eiωj ) Im
×
sym


(k,`)
j=1
k=−∞ `=−∞
ω (3)

(k) 2(1−H)
∞
ω
X
−1
1
−3i
 (k) (`) (k+`)

c0 σ04
(1 − eiωj )
,
=
 ω1 ω2 ω3
sym
(`) (k+`) 
2π sin(Hπ)Γ(3 − 2H)
(k,`)
ω2 ω3
ω (3)
j=1
k=−∞ `=−∞
3
Y

∞
X
(3.94)
which is finite for every ω (2) ∈ (0, 2π)2 such that ω1 ω2 ω3 6= 0, because it is even integrable on [0, 2π]2 , as
we will see immediately. In (3.94) changing the order of the limit and the infinite sum can be justified by
the Beppo Levi theorem and the fact that the convergence (3.92) is essentially monotone. Therefore let us
define the function
(3.95)
S3,∆J ω (2) $ lim S3,∆Jλ ω (2) ,
λ→0
71
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
for every ω(2) ∈ (0, 2π)2 such that ω1 ω2 ω3 6= 0. Then S3,∆J ω(2)
we have
Z
Z
S3,∆J ω(2) dω(2) =
[0,2π]2
Z
6 lim
λ→0
[0,2π]2
= lim µ0 λ3−2H σ04
λ→0
is integrable since by the Fatou lemma
lim S3,∆Jλ ω(2) dω(2)
[0,2π]2
λ→0
S3,∆Jλ ω(2) dω(2) = lim cum (∆Jλ (0), ∆Jλ (0), ∆Jλ (0))
λ→0
3
(1 − H)(2H − 1)H
= c0 σ04
2
2H+1
2H
λ2H+1 − (λ + 1)
+ (λ + 1) + λ2H
2H + 1
3
< ∞.
(1 − H) H(2H + 1)
(3.96)
The third equation in (3.96) is the consequence of formula (3.83). Moreover,
Z
[0,2π]2
Z
ei(t1 ω1 +t2 ω2 ) S3,∆J ω(2) dω (2) =
Z
= lim
λ→0
[0,2π]2
ei(t1 ω1 +t2 ω2 ) lim S3,∆Jλ ω(2) dω (2)
λ→0
[0,2π]2
ei(t1 ω1 +t2 ω2 ) S3,∆Jλ ω (2) dω(2) = lim cum (∆Jλ (0), ∆Jλ (t1 ), ∆Jλ (t2 ))
λ→0
= cum (∆J(0), ∆J(t1 ), ∆J(t2 )) ,
(3.97)
where the second equation can be justified by applying the Beppo Levi theorem both to the real and to the
imaginary parts, and considering that the convergence in(3.95)
is essentially monotone. The last equation
in (3.97) is the consequence of Theorem 6. Thus S3,∆J ω (2)
is really the bispectrum of the ∆LISDLG
process ∆J(t). Equation
(3.91)
follows from (3.95) and (3.94).
The fact that S3,∆J ω (2) is real-valued and positive, is the consequence of the fact that the same property
−1
3
Q
(k,`)
(k) (`) (k+`)
(1 − eiωj ) ω1 ω2 ω3
on the right-hand side of
and (−i)
holds for each of the terms S3,J 0 ω(2)
j=1
the first row of (3.91).
Remark 20. The bispectrum S3,∆J ω (2) behaves around zero as
S3,∆J (ω1 , ω2 )
'
ω1 ,ω2 →0
q
−2H
2
2
ω1 + ω2
.
Remark 21. The property of the bispectrum S3,∆J ω (2) that it is real-valued is equivalent to the property
of the cumulants that
cum (∆J(0), ∆J(t1 ), ∆J(t2 )) = cum (∆J(0), ∆J(t2 − t1 ), ∆J(t2 )) ,
t1 , t2 ∈ Z, see also Remark 13.
Proof. See the proof of Remark 17.
(3.98)
72
E. IGLÓI AND C. TERDIK
On the other hand, the generalization of (3.98) for cumulants of order m > 3 also holds, that is to say,
we have
Remark 22.
cum (∆J(0), ∆J(t1 ), . . . , ∆J(tm−1 )) = cum (∆J(0), ∆J(tm−1 − tm−2 ), . . . , ∆J(tm−1 )) ,
for all m ≥ 3 and t1 , . . . , tm−1 ∈ Z.
Proof. Similar to that of Remark 18.
3.4. Nonlinearity of the ∆ISDLG and ∆LISDLG processes
The notion of nonlinearity has several meanings in time series analysis. The most frequent is the following.
Let X(t), t ∈ Z, be a discrete time L2 -stationary process with zero mean. It is called a linear process, if
it has an L2 -convergent moving average representation with independent and identically distributed (i.i.d.)
input variables, i.e.
∞
X
X(t) =
c(k)Z(t − k),
(3.99)
k=−∞
where
∞
X
c(k)2 < ∞,
(3.100)
k=−∞
and the input random variables Z(t), t ∈ Z, are i.i.d., E Z(t) = 0 and E Z(t)2 < ∞. Otherwise process X(t)
is said to be nonlinear.
Let us consider a third order stationary process X(t), linear in the above sense and having bispectrum
(in the sense of the paragraph before Prop. 23). By the Marcinkiewicz–Zygmund inequality (see e.g. [35])
the input random variables Z(t) also have finite third order moment. Therefore the bispectrum of process
X(t) has the form
cum (Z(0))
3
ϕ(ω1 )ϕ(ω2 )ϕ(−ω1 − ω2 ),
S3,X ω (2) =
2
(2π)
where
ϕ(ω) =
∞
X
c(k)e−ikω ,
k=−∞
ω ∈ [0, 2π), is the frequency domain transfer function of process X(t) with moving average representation
(3.99), see [15], Lemma 7. However, if X(t) is also linearly regular and long-range dependent then such a
bispectrum cannot be real. Summing up, it holds true the following lemma, which is a somewhat sharpened
version of the theorem in [15].
Lemma 6. If a discrete time process with zero mean is linearly regular, long-range dependent and it has
real-valued bispectrum, then it is nonlinear.
The discrete time processes ∆Jλ (t) and ∆J(t) are linearly regular, because the spectrum of each process
obviously satisfies the Kolmogorov–Szegő condition. They are long-range dependent as well. Taking into
account that they have real-valued bispectra, see Propositions 24 and 10, we can derive the following theorem.
Theorem 11. Both the ∆ISDLG process ∆Jλ (t) and the ∆LISDLG process ∆J(t) are nonlinear.
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
73
3.5. Relation with [1], a similar result of Barndorff–Nielsen
A paper [1] by Barndorff–Nielsen also deals with the superposition of OU processes and the limit of
its integral process. The buildup of its model is very similar to ours, but the terminology and the initial
processes, i.e. the processes to be superposed are totally different. In effect the treatment in [1] is somewhat
more general than ours, so we will test it only from our point of view.
Our notation is unlike that of the paper under review in order to make the correspondence of the two
models evident. Now, in [1] there is defined a family Y (t, ·), t ∈ R, of random measures Y (t, B), B ∈ B(R+ ),
on R+ , in terms of integrating with respect to an independently scattered random measure Z(dt, dx) on the
set R × R+ , i.e.
Z
Y (t, B) $
e
B
(x)
−xt
Zxt
es Z(ds, dx),
(3.101)
−∞
(s)
and the process proper is
Y (t) $ Y (t, R+ ).
(3.102)
From (3.101) and (3.102) one has
Z
(−xY (t, dx)dt + Z(xds, dx)) ,
dY (t) =
R+
(x)
showing that Y (t) is really a superposition of perhaps infinitesimally determined OU processes.
Let the generalized Lévy measure Q(A, dt) in the Lévy–Khinchin representation
log ϕZ(A) (u) = log E eiZ(A)u =
Z
eiut − 1 − iuτ (t) Q(A, dt),
R
where A ∈ B (R × R+ ), factorize as
Q(A, dt) = M (A)V (dt),
(3.103)
for some measure M on R × R+ and some Lévy measure V on R. Moreover let M factorize as
M (ds, dx) = dsN (dx),
(3.104)
for the Lebesgue measure ds on R and a probability measure N on R+ . The paper under discussion, similarly
to our paper, uses the Γ distribution, but with a different shape parameter, namely,
N $ Γ(2(1 − H), 1/λ),
(3.105)
where the parameters are 1/2 < H < 1 and 0 < λ. We denote the dependence on λ as we did in the case of
the SDLG process Y (t) = Yλ (t).
The joint cumulants of the superposition process Y (t), if they exist, are also given in the formula (3.5)
by [1], showing that
Z
cum (Y (t1 ), . . . Y (tm )) =
R+
cum Y (x) (t1 ), . . . Y (x) (tm ) N (dx),
(3.106)
74
E. IGLÓI AND C. TERDIK
i.e. the mixture of the corresponding cumulants of the component processes Y (x) (t), x ∈ R+ , by the weight
distribution N . For each x ∈ R+ , the component process Y (x) (t) in (3.106) is the OU process defined by the
Langevin equation
dY (x) (t) = −xY (x) (t)dt + Z (x) (xdt),
where the independently scattered random measure Z (x) (dt) $ Z(dt, dx), choosing the measure N in (3.104)
to be
N = Px ,
(3.107)
the atomic measure concentrated at point x. That is, Y (x) (t) and Z (x) (dt) are the same as Y (t) and
Z(dt, dx), respectively, except that N is different, being defined by (3.107) and (3.105). We will also denote
(x)
the dependence on λ as, e.g., Yλ (t).
The integrated process
Zt
Jλ (t) $
Yλ (s)ds
0
is also defined, and a theorem is stated and proved about the convergence as λ → 0, of the family of processes
Jλ (t), λ > 0. More precisely, assume that the Lévy measure V in (3.103) also depends on λ, viz.
Vλ (dt) = λ2(H−1) V (dt).
(3.108)
Under condition (3.108) and some other condition imposed on the logarithmic characteristic function of
the stationary process Y1 (t), the finite-dimensional distributions of Jλ (t) converge weakly to those of some
process J(t), i.e.
w
Jλ −→ J
λ→0
in the finite-dimensional sense, see Theorem 6.1 of [1]. Of course, process J(t) has stationary increments.
Now, assuming that the Lévy measure V is such that J(1) has a finite second order moment, we have
2H
2H
2H
(3.109)
cov (J(t1 ), J(t2 )) = const. |t1 | + |t2 | − |t2 − t1 |
(see Ex. 6.2 in [1]), the same as the covariance of the LISDLG process J(t) (and of the FBM). More generally,
there is a close resemblance between the two processes J(t). In particular, as long as the cumulants exist,
for both of them
cumm (J(t)) = k(m)tm+2(H−1)
holds for m > 2, where k(m), m > 2, are some positive constants not depending on t, see Theorem 7 in this
paper and Example 6.3 in [1]. (Naturally, constants k(m) are different for the two processes J(t).) Thus,
both processes J(t) are multifractal with the same exponents.
It follows from (3.109) that the generalized second order spectra appertaining to the two processes are
1−2H
, see Proposition 20. We state this in the following proposition:
the same, namely, S2,J 0 (ω) =const.|ω|
Proposition 25. The generalized spectrum appertaining to process J(t) of [1], i.e.
S2,J 0 (ω) $ lim S2,Yλ (ω) ,
λ→0
is given by
1−2H
S2,J 0 (ω) = const. |ω|
.
75
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
Thus it is the same as the generalized spectrum appertaining to the LISDLG process J(t).
After all, the two processes are significantly different. The discrepancy shows itself by means of the
different properties of the generalized bispectra. That is to say, the generalized bispectrum appertaining
to the LISDLG process J(t) is real-valued, see Proposition 22, in contrast to the generalized bispectrum
appertaining to process J(t) of [1].
Proposition 26. The generalized bispectrum appertaining to process J(t) of [1], i.e.
S3,J 0 ω (2) $ lim S3,Yλ ω(2) ,
λ→0
if it exists at all, cannot be real-valued apart from the case when it is zero.
(x)
Proof. We may assume that the Lévy measure V in (3.108) is such that the random variables Zλ (1) $
(x)
Zλ ((0, 1]), x ∈ R+ , λ > 0, have finite third order cumulants, because in the opposite case the bispectra of
(x)
processes Yλ (t), x ∈ R+ , λ > 0, do not exist, thus, neither do the bispectra of processes Yλ (t), λ > 0, so
the generalized bispectrum appertaining to process J(t) is not defined either.
(x)
Since for each x ∈ R+ process Yλ (t) is an OU process with a zero expectation and a finite second order
moment, its spectral representation is
Z
(x)
eitω
Yλ (t) =
R
1
(x)
W (dω),
iω + x λ
(x)
(x)
with some stochastic spectral measure Wλ (dω) on R. Since Yλ (t) is a linear process with an integrable
transfer function in its moving average representation (being an OU process), its bispectrum is the three-term
product
1 2
1
1
1
·
·
c3 (x, λ)
S3,Y (x) ω (2) =
λ
2π
iω1 + x iω2 + x −i(ω1 + ω2 ) + x
=
1
2π
2
c3 (x, λ)
x x2 + ω12 + ω22 + ω1 ω2 − i(ω1 + ω2 )ω1 ω2
2
2
x2 (x2 + ω12 + ω22 + ω1 ω2 ) + (ω1 + ω2 ) ω12 ω22
(3.110)
(see Ex. 3 in [3]), where
(x)
(x)
c3 (x, λ) = x cum3 Zλ (1) = xλ2(H−1) cum3 (Z1 (1)) = xλ2(H−1) ce3 ,
(3.111)
and ce3 ∈ R is a constant depending neither on x nor on λ.
(x)
On the other hand, because of (3.106) the bispectrum of Yλ (t) is the mixture of the bispectra of Yλ (t),
i.e.
Z
S3,Y (x) ω (2) N (dx),
(3.112)
S3,Yλ ω (2) =
λ
R+
where the distribution N is the Γ distribution given by (3.105). Hence by substituting the results of (3.110)
and (3.111) into (3.112), one has
S3,Yλ ω(2) =
1
2π
2
Z
2(H−1)
λ
ce3
R+
x
x x2 + ω12 + ω22 + ω1 ω2 − i(ω1 + ω2 )ω1 ω2
2
2
x2 (x2 + ω12 + ω22 + ω1 ω2 ) + (ω1 + ω2 ) ω12 ω22
N (dx),
76
E. IGLÓI AND C. TERDIK
and therefore

S3,J 0 ω(2) =
ce3

lim λ2(H−1)
2 λ→0
(2π)
λ2(1−H)
Γ(2(1 − H))
Z
x2(1−H) e−λx
R+

x x + ω12 + ω22 + ω1 ω2 − i(ω1 + ω2 )ω1 ω2

×
dx
2
2
x2 (x2 + ω12 + ω22 + ω1 ω2 ) + (ω1 + ω2 ) ω12 ω22
2
= ce3
1
2
(2π) Γ(2(1 − H))
− ice3
Z
x3−2H x2 + ω12 + ω22 + ω1 ω2
2
R+
(ω1 + ω2 )ω1 ω2
2
x2 (x2 + ω12 + ω22 + ω1 ω2 ) + (ω1 + ω2 ) ω12 ω22
Z
2
(2π) Γ(2(1 − H))
x2(1−H)
2
R+
dx
x2 (x2 + ω12 + ω22 + ω1 ω2 ) + (ω1 + ω2 )2 ω12 ω22
dx.
Now, if the generalized bispectrum S3,J 0 ω (2) is real, then ce3 = 0, thus S3,J 0 ω (2) = 0.
4. Application to computer network traffic data
The earlier paper [41] raised the problem of finding a stochastic process all of whose increment processes
have the following properties:
•
•
•
•
•
stationarity;
LRD;
real-valued bispectrum;
nonlinearity;
Γ one-dimensional distribution (or nearly Γ in some sense).
Multifractal property was not positively required but was alluded to. The motivation was that the computer
network traffic data set investigated by the authors of that paper mainly from the point of view of spectralbispectral analysis, showed basically the above properties. The ∆LISDLG process does have these properties.
Here we wish to demonstrate, in particular graphically, that a fair fit can be found between the network
traffic data centralized to zero and the ∆LISDLG process.
We analyzed the time series of aggregated 155 Mbps SUNET ATM WAN (Swedish University Network,
Asynchronous Transfer Mode, Wide Area Network) traffic3 on the ≈0.1 s (approximately 0.1 s) level. That
is to say, the source data set consisting of 425 million cell arrivals (slots) was divided into blocks of size
8000 slots, approximately this is the amount of slots piping in 0.1 s. Then for each block the block sum was
calculated. Thus the aggregated data set consists of about 425 million/8000 ≈ 53000 measurements. It will
be called ≈0.1 s level ATM data, or briefly ATM data. We will also use the data set called ≈1 s level ATM
data arising by aggregating the ≈0.1 s level ATM data in each block of size 10.
4.1. The dilative stability and parameter estimation
At first we demonstrate one-dimensional dilative stability, a particular manifestation of the multifractal
property, see Proposition 17. Figure 1 shows in logarithmic scale of how the estimated cumulants of the
aggregated data depend on the level of aggregation. The aggregation is meant in the sense of Corollary 4,
i.e. it means averaging. The orders of the cumulants is in the range {2, . . . , 6}. If the data came from a
3 The reported ATM WAN measurements were carried out by the CARAT laboratory of Telia Research in Sweden. We would
like to thank them and Sándor Molnár in High Speed Networks Lab., Dept. of Telecommunications & Telematics, Technical
University of Budapest, for providing the data. Special thanks go to our colleague Zoltán Gál in Center for Informatics and
Computing, University of Debrecen, for joint work.
77
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
∆LISDLG process, then by taking the logarithm in (3.34) for each fixed order m ∈ {2, . . . , 6} the relationship
would be
(4.1)
log cumm ∆J (ni ) = log k1 (m) + 2(H − 1) log ni ,
where the levels of time-aggregation are ni , i = 1, . . . , r. In Figure 1, r = 25 and ni ranges from n1 = 4 to
n25 = 223 such that log ni , i = 1, . . . , 25, is an approximately equidistant sequence. For each cumulant order
m, (4.1) means that the dependence is linear and the lines must be parallel, each with slope 2(H − 1). This is
so in Figure 1, where the five linear point sets, corresponding upwards to m = 2, . . . , 6, confirm the ∆LISDLG
model. The common slope parameter 2(H − 1) and constants log k1 (m), m = 2, . . . , 6, were estimated by
weighted linear regression from the linear model with equations (4.1), i = 1, . . . , r, m = 2, . . . , 6, where the
weights are the inverse of the standard deviations of the logarithm of the estimated cumulants. The LRD
parameter H estimated in this way came out to be4
b = 0.9156.
H
(4.2)
In possession of the estimated H, the constants depending on m and H in (3.34), including the integral term
could be obtained. By the way, the latter term was computed by Monte-Carlo method. Then constants c0
c2 = 29.0833.
and σ02 were estimated, also by linear regression, to be cb0 = 38.3156 and σ
0
30
log(estimated cumulant)
25
20
15
10
5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
log(n)
Figure 1. One-dimensional dilative stability illustrated by cumulants. The discrete
points corresponds to the
data. Points plotted with the same signs have coordinates
(ni )
\
∆J
, i = 1, . . . , 25. The orders of the cumulants are m = 2, . . . , 6,
log ni , log cum
m
upwards. The continuous lines fitted by weighted linear regression have the common slope
b − 1), where H
b = 0.9156.
2(H
4 Examination
of statistical methods is not the aim of this paper. However, it should be noticed that it is not hard to see
that stationarity and the law of large numbers ensure the consistency of the cumulant estimates and the characteristic function
estimates, and in this wise the consistency of the estimators used in this section.
78
E. IGLÓI AND C. TERDIK
In place of the cumulants the characteristic function is also available to demonstrate one-dimensional
dilative stability and to estimate the LRD parameter H. Taking the logarithm twice and then the real part
in (3.44), we have
Re log log ϕ J(t) (u) = 2(H − 1) log t + Re log log ϕJ(1) (u).
t
Changing to the notation of Corollary 4, this means that
Re log log ϕ∆J (ni ) (u) = 2(H − 1) log ni + Re log log ϕ∆J (1) (u).
(4.3)
When testing the ATM data from this point of view, we used the same range of the levels of aggregation ni
as above. The arguments u of the estimated characteristic functions were 100 points equally spaced in the
interval (0, 0.05]. (The reason for the choice of 0.05 as the end point is that the variance of the estimated
characteristic function and also the variance of the curves increase apace for larger values of u.) Figure 2
plots for each fixed level of aggregation ni the left-hand side of (4.3) as a function of u. If the ATM data
set were a realization of the ∆LISDLG process, then by (4.3) these curves should be approximately parallel.
And it seems they really are.
Figure 2. One-dimensional dilative stability. Each curve is the graph of a function
Re log log ϕ
b∆J (ni ) (u) as function of u. The aggregational levels ni are the same as in Figure 1.
The curves are approximately parallel.
Figure 3 plots for different values of u the left-hand side of (4.3) as a function of log ni . If the ATM data
set were a realization of the ∆LISDLG process, then by (4.3) these curves should be approximately parallel
lines, each with the approximate slope 2(H − 1). And it seems they really are.
If the data come from a ∆LISDLG process, the previous method can be used to estimate parameter H
from the linear model (4.3). We also tried it for wider ranges of the arguments of characteristic function.
Figure 4 shows the result when 500 equidistant values of uj in the interval (0, 10) are used. The mean of the
b j ), j= 1, . . . , 500, is
500 estimated values H(u
b = 0.9452.
H
(4.4)
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
79
Figure 3. One-dimensional dilative stability. Each line is the graph of a function
Re log log ϕ
b∆J (ni ) (u) as function of log ni . The aggregational levels ni are the same as in
Figures 1 and 2. The values of u are the same as in Figure 2. Each line has approximately
the same slope 2(H − 1).
b j ) remains similar
If even larger intervals are used in place of (0, 10), the arrangement of the estimations H(u
to the one in Figure 4. Thus, we have good reason for supposing the value in (4.4) to be an adequate
estimation of parameter H.
4.2. Fitting of distributions and spectra
4.2.1. Fitting of the one-dimensional distributions
Figure 5 shows the histogram of the ATM data centralized to zero and the density function of the shifted
Γ distribution fitted by the method of moments using the second and third moments. Though it is known
that the distribution of the ∆LISDLG process ∆J(t) is not exactly shifted Γ, it is close to it if the LRD
parameter H is near to 1, see Remark 14. We do not know how rapid the convergence is; anyhow it seems
b = 0.9452 in (4.4) is sufficiently close to 1, the distribution of ∆J(t) being near
that the estimated value H
to the shifted Γ.
Let us now examine what the fitting looks like for characteristic functions. We know the characteristic
function of ∆J(1) only in the interval −σ0−2 , σ0−2 , where the series expansion of its logarithm holds, see
Proposition 15. Using the value (4.4) for H, we searched for the values of the other two parameters c0 and σ02
giving the best least squares fitting, in the interval −0.9σ0−2 , 0.9σ0−2 , between the characteristic function
b∆J(1) (u), the latter being estimated from the ATM data. (The coefficient
ϕ∆J(1) (u) and the empirical one ϕ
0.9 is used because the series expansion begins to diverge, or rather, to converge slowly near the end points
of the convergence interval. This effect occurs even for the series expansion of the log-characteristic function
of the Γ distribution.) The search ended on finding the optimal values
cb0 = 12.7201
c2 = 62.1118,
σ
0
(4.5)
80
E. IGLÓI AND C. TERDIK
Figure 4. Linear
estimators of H based on the linear model (4.3). The points
regression
b
have coordinates uj , H(uj ) , where uj , j= 1, . . . , 500, are equidistant in the interval (0, 10),
b j ) is the linear regression estimator based on (4.3) when the range of the aggregaand H(u
tional levels ni is the same as in Figure 1.
and the best fitting can be seen in Figure 6. The value giving the end points of the interval used at the final
−1
c2 = 0.0145. The curve consisting of discrete points is the estimated characteristic function
stage is 0.9σ
0
ϕ
b∆J(1) (u), and the best fitting ϕ∆J(1) (u) is the continuous curve. Also the best fitting Γ characteristic
function is plotted, but it is not shown as it is overlapped by the other optimal curve.
4.2.2. Fitting of the spectra and bispectra
To demonstrate the fitting of the theoretical spectra of the ∆LISDLG process to the empirical spectra,
we use the ≈1 s level ATM data because the coincidence of the second order spectra is rather convincing at
this level of aggregation. Thus, the level of aggregation of the ∆LISDLG process too has to be changed by a
factor of 10. Or rather, we can get the same result changing not the level of aggregation, but parameters c0
and σ02 , according to the scaling behavior (3.33) or (3.34) of the cumulants. For this purpose, consider the
parameters of the ∆LISDLG process fitted above to the ≈ 0.1 s level ATM data, i.e.
H = 0.9452
c0 = 12.7201
σ02 = 62.1118,
see (4.4) and (4.5). Now the parameters of the ∆LISDLG process, after changing the time scale by a factor
of 10, are
H = 0.9452
c0 = 12.7201 · 102(H−1) = 9.883
σ02 = 62.1118.
(4.6)
81
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
Figure 5. Histogram of the ATM data centralized to zero and the shifted Γ density fitted to them.
Thus when computing the theoretical spectrum and bispectrum by the corresponding formulas (3.76) and
(3.91), respectively, we used the parameters in (4.6).
Figure 7 shows the fitting of the estimated spectrum alias smoothed periodogram and the theoretical
spectrum (3.76) of the ∆LISDLG process. More precisely, in Figure 7 both spectra are normed so that their
geometric mean be equal to 1.
Figure 8 shows is the theoretical bispectrum. Figures 9 and 10 show the real part and the imaginary part,
respectively, of the estimated bispectrum. More precisely, in Figures 8, 9 and 10 both the theoretical and the
empirical bispectrum are normed so that the arithmetic mean of their real parts be equal to 1. Now observe
the imaginary part of the estimated bispectrum (Fig. 10) fluctuating around zero with a small amplitude as
compared to the range of the real part (Fig. 9). Thus, the imaginary part seems to be negligible. This fits
in with the fact that the imaginary part of the theoretical bispectrum is zero. Also note the similarity of the
real parts, see Figures 8 and 9.
5. Open problems and directions of future research
It seems to be a difficult task to calculate
Z
Dτ0 (s)2(H−1) ds
(5.1)
[0,1]m
in closed form, in order to get the form of the dependence on dimension m. The difficulty arises from the
fact that for m > 4 the term Dτ0 (s) is not symmetric in its variables, hence for different permutations τ the
integrals on the simplexes {s ∈ [0, 1]m : sτ1 6 . . . 6 sτm } are different as a rule. This is a relevant problem
because the m-th cumulant of the distribution of the LISDLG process J(t) differs in the factor (5.1) from
the m-th cumulant of the corresponding Γ distribution, see (3.33).
A problem connected with the previous one is to specify the distribution of the LISDLG process J(t),
and hence that of the ∆LISDLG process ∆J(t) (for fixed t, naturally). Because of the one-dimensional
dilative stability, see Proposition 17, it would suffice to give the distribution of J(1). The authors know the
82
E. IGLÓI AND C. TERDIK
Figure 6. Real (left) and imaginary (right) parts of the following three characteristic
functions: the empirical one estimated from ATM data (discrete points); the best fitting
∆LISDLG and the best fitting Γ characteristic fuction (both plotted with continuous lines
and exactly overlapped by each other). When fitting the ∆LISDLG characteristic function,
parameter H was fixed at the value given by (4.4). The parameters yielding the best fitting
characteristic function ϕ∆J(1) (u) are the ones in (4.5). The end points of the interval used
c2 −1 , which are ±0.0145 at the final stage.
for the optimizing procedure are ±0.9σ
0
series expansion of the log characteristic function only in a certain neighborhood of zero, see Proposition 15.
Reference [10] gives an elegant closed-form expression for the whole characteristic function of the integral
on [0, t], of a squared OU process. Using it as well as the mixture representation of the logarithm of the
characteristic function, one obtains
µ0 λ3−2H
log ϕJλ (1) (u) =
Γ(3 − 2H)σ02
Z∞
2(1−H) −λx
x
e
−iuσ02
+ g(u, x) dx,
x
0
where
g(u, x) $
x
x
1
1
1
1
− log
z (u, x) + cosh
z (u, x) ,
z (u, x) +
sinh
2 x
2
z (u, x)
2
2
and
r
z (u, x) =
1−
4iuσ02
·
x
Taking limit in (5.2) the characteristic function ϕJ(1) (u), u ∈ R, could be derived, since
log ϕJ(1) (u) = lim log ϕJλ (1) (u),
λ→0
(5.2)
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
83
Figure 7. The smoothed periodogram of the ≈1 s level ATM data and the theoretical
spectrum (3.76) of the ∆LISDLG process with parameters as in (4.6). Both curves are
normed so that their geometric mean be 1.
u ∈ R, if the function
x 7−→ x
2(1−H)
−iuσ02
+ g(u, x) ,
x
x ∈ R+ , proved to be integrable5 .
Note also that expression (5.1) in closed form would be useful for deriving the Fourier transform of the
Lévy measure in the Lévy–Khinchin representation of log ϕJ(1) 6 .
Concerning the spectra of order m > 4, it follows easily from Definition 9 and Remark 17 that they must
be real provided they exist, similarly to the case m = 3, i.e. the case of the bispectrum. The question is
whether they exist, and if they do, is there a general form of the m-th order spectrum as a function of m
too? This problem is at least as hard as to compute (5.1) in closed form, because it needs the knowledge of
the m-th order spectrum of the DLG process. The latter however makes it necessary to calculate an integral,
similar to but even more complicated than, equation (5.1), see the proof of Proposition 9. Anyway, the
authors know the closed expression (3.37) for the cumulants of the derivative of the LISDLG process J(t).
Though the derivative of the LISDLG process J(t) is only an informal thing, its cumulants can be defined
exactly. Is there a closed form expression, similar to (3.37), for cum (∆J(0), ∆J(t1 ), . . . , ∆J(tm−1 )) which
also holds for all m > 4?
Another direction for future research is the fitting of ∆LISDLG processes to simulated and real data.
First of all, concerning the simulation it must be noted that there is a very effective method to simulate a
DLG process in discrete time points, see Remark 4. Probably, one can put this method to use for simulating
5 After
u∈
submitting the first version of this paper, the authors pushed through this idea, so the cumulant function log ϕJ (1) (u),
R, has been obtained since then.
6 The
referee was so kind to draw the authors attention to this fact.
84
E. IGLÓI AND C. TERDIK
Figure 8. The theoretical bispectrum (3.91) of the ∆LISDLG process with parameters as
in (4.6). It is normed so that its arithmetic mean be 1. It is plotted on the determining
triangle with vertices (0, 0), (π, 0), (2π/3, 2π/3), see [37] (p. 29).
∆LISDLG process7 . Secondly, adapting existing parameter estimation methods and tests of model fitting
to our process is a promising task, as well as contriving and investigating new ones, including those we used
in Section 4.
Stochastic modelling of computer network data traffic flow processes using the ∆LISDLG process is
another topic for future research. The motivation comes not only from the apparently convincing fitting
found by using various graphical methods and several data sets, but also from the existing realistic models
for the DLG processes which are the component processes for our superposition step. In particular, the
DLG process is a CBI process with a.s. continuous trajectories, see Theorem 1, and the latter process can
be represented as a scaled limit of Galton–Watson processes with immigration, see Theorem 2.2 in [20].
Another practical model is a Cox process with DLG process intensity, resulting in a birth and death process
with immigration, see [7].
Appendix A. Multivariate Γ distribution
The accompanying Gaussian distribution
Let m ∈ N, and let Σ = (σij )i,j=1,... ,m be an m × m matrix with the following property:
Σ is symmetric, positive definite and σ11 = . . . = σmm = σ 2 .
(A.1)
Let X = (X1 , X2 , . . . , Xm ) be a zero mean Gaussian vector with covariance matrix Σ. Define Y = (Y1 , Y2 ,
. . . , Ym ) as the random vector with entries Yk = Xk2 , k = 1, 2, . . . , m. The characteristic function of Y is
−1/2
ϕY (u) = |I − 2iU Σ|
,
7 After submitting the first version of this paper, the authors pushed through this idea. More specifically, processes approximating the ∆LISDLG process with adequate precision have been simulated.
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
85
Figure 9. The real part of the estimated bispectrum of the ≈1 s level ATM data. It is
normed so that its arithmetic mean be equal to 1. It is plotted on the same triangle as in
Figure 8.
diagonal, which is u,
where U $ diag(u) with u ∈ Rm , i.e. a matrix with zero elements except for the mainP
I is the m × m identity matrix, and | | denotes the determinant. If one takes Z = nk=1 Yk , the sum of
n ∈ N independent copies of the vector Y, then one gets the characteristic function
−n/2
ϕZ (u) = |I − 2iU Σ|
.
(A.2)
All the marginals of ϕZ (u) are χ2 distributed with degree of freedom n which is a particular case of the Γ
distribution. The Γ (α, β) distribution is defined in general either by the density
fΓ(α,β) (x) =
xα−1
exp (−x/β) , if x > 0 and zero otherwise,
β α Γ (α)
or by the characteristic function
−α
ϕΓ(α,β) (u) = (1 − iuβ)
.
Its cumulants are
cumm (Γ (α, β)) = (m − 1)!αβ m ,
m = 1, 2, 3, . . . Parameters α > 0 and β > 0 are called the shape
parameter and the scale parameter,
respectively. In that context the marginals of ϕZ (u) are Γ n/2, 2σ 2 . The entries of Y are correlated and
χ2 distributed with degree of freedom 1. The cumulants of Y are the same as the cumulants of Y − σ2 =
(H2 (X1 ) , H2 (X2 ) , . . . , H2 (Xm )) , where H2 (Xk ) = Xk2 − σ 2 is the Hermite polynomial of second order.
The cumulants of Hermite polynomials are well-known, in particular in the following case.
86
E. IGLÓI AND C. TERDIK
Figure 10. The imaginary part of the estimated bispectrum of the ≈1 s level ATM data. It
is normed by the same constant with which the real part. It is plotted on the same triangle
as in Figure 8.
Proposition 27.
cum (H2 (X1 ) , H2 (X2 ) , ..., H2 (Xm )) =
X
cov (X1 , Xi1 ) cov (Xi1 , Xi2 ) · · · cov Xim−2 , Xim−1 cov Xim−1 , X1 ,
2m−1
(i1 ,... ,im−1 )
∈Perm(2,... ,m)
see [40], Example 10, for details.
E.g., if m = 2, one can see that
2
,
cum (H2 (X1 ) , H2 (X2 )) = cov (H2 (X1 ) , H2 (X2 )) = 2σ12
and if m = 3,
cum (H2 (X1 ) , H2 (X2 ) , H2 (X3 )) = 8σ12 σ23 σ31 .
Although one cannot change the exponent n/2 in (A.2) for an arbitrary positive real number α, because the
result might not be a characteristic function, see [24], we define the function
−α
ϕ(u) = |I − 2iU Σ|
,
(A.3)
for further consideration.
Definition 11. If function ϕ(u) defined by (A.3) is a characteristic function, then the m-dimensional Γ
distribution with shape parameter α and scale parameter matrix 2Σ according to the accompanying Gaussian
distribution, Γ (α, 2Σ) for short, is defined, in particular by its characteristic function ϕ(u).
SUPERPOSITION OF DIFFUSIONS AND ITS MULTIFRACTAL LIMIT
87
Let Z = (Z1 , . . . , Zm ) be Γ (α, 2Σ) distributed. It is easy to see that each component of Z has onedimensional Γ distribution, namely,
Zk ∼ Γ α, 2σ 2 ,
k = 1, . . . , m. Moreover, for 1 6 k 6 m, each subset (Zi1 , . . . , Zik ) also has k-dimensional Γ distribution.
Naturally, there are other m-dimensional distributions whose one-dimensional marginals are Γ distributions,
however the one given above is the most plausible of them, because of the following:
Proposition 28 (see Th. 1 in [24]). The distribution Γ (n/2, 2Σ) is just the marginal distribution of the
diagonal elements of a Wishart matrix.
For these special values of the shape parameter, i.e. for half-integer positive α, and arbitrary Σ with
property (A.1), the multi-dimensional Γ (α, 2Σ) distribution was defined in [25]. Reference [24] draws our
attention to the fact that for m > 2, if one makes no restraints for the parameters, other than α > 0 and Σ
fulfils (A.1), then (A.3) is not necessarily a characteristic function. However, the infinite divisibility of the
Γ (1/2, 2Σ) distribution is a necessary and sufficient condition for the Γ (α, 2Σ) distribution to exist for all
α > 0. Reference [14] gave a sufficient condition for infinite divisibility, namely that Σ is the autocovariance
matrix of a Gauss–Markov vector. The latter condition comes true for the very case of an autocovariance
matrix of a stationary OU process. To conclude, we state the following:
Proposition 29 (see [14]). If α > 0 and Σ is the autocovariance matrix of the vector (X(t1 ), . . . , X(tm )),
where t1 < . . . < tm and X(t) is a stationary OU process, then the Γ (α, 2Σ) distribution really exists.
The cumulants of distribution Γ (α, 2Σ) are the same as the cumulants of Γ (n/2, 2Σ) but for some multiplicative constant because the cumulant generating function is log ϕ(u). Let Z = (Z1 , . . . , Zm ) be Γ (α, 2Σ)
distributed, then
cum (Z1 , Z2 , ..., Zm ) = 2α cum (H2 (X1 ) , H2 (X2 ) , ..., H2 (Xm ))
X
σ1i1 σi1 i2 . . . σim−1 1 ,
= 2m α
(A.4)
(i1 ,... ,im−1 )∈Perm(2,... ,m)
where X = (X1 , X2 , . . . , Xm ) is the accompanying Gaussian random vector. The cumulants (A.4) are invariant under changing the signs of the X’s, therefore the distribution Γ (α, 2Σ) does not define the accompanying
Gaussian distribution uniquely.
The authors are grateful to the referee for the careful review and the constructive attitude.
References
[1] O.E. Barndorff–Nielsen, Superposition of Ornstein–Uhlenbeck type processes, Technical Report 1999-2, MaPhySto. Aarhus
University (1999), URL: http://www.maphysto.dk/cgi-bin/w3-msql/publications/.
[2] J. Beran, Statistics for Long-Memory Processes. Chapman & Hall, Monogr. Statist. Appl. Probab. 61 (1994).
[3] D.R. Brillinger, An introduction to polyspectra. Ann. Math. Statist. 36 (1965) 1351-1374.
[4] L. Calvet and A. Fisher, Forecasting multifractal volatility. J. Econometrics 105 (2001) 27-58.
[5] P. Carmona and L. Coutin, Fractional Brownian motion and the Markov property, Electron. Commun. Probab. 3 (1998)
95-107.
[6] P. Carmona, L. Coutin and G. Montseny, Applications of a representation of long-memory Gaussian processes. Stochastic
Process. Appl. (submitted), URL: http://www.sv.cict.fr/lsp/Carmona/prepublications.html
[7] P. Clifford and G. Wei, The equivalence of the Cox process with squared radial Ornstein–Uhlenbeck intensity and the
death process in a simple population model. Ann. Appl. Probab. 3 (1993) 863-873.
[8] D.R. Cox, Long-range dependence, non-linearity and time irreversibility. J. Time Ser. Anal. 12 (1991) 329-335.
[9] J.C. Cox, J.E. Ingersoll and S.A. Ross, A theory of the term structure of interest rates. Econometrica 53 (1985) 385-407.
[10] T. Dankel Jr., On the distribution of the integrated square of the Ornstein–Uhlenbeck process. SIAM J. Appl. Math. 51
(1991) 568-574.
[11] E.B. Dynkin, Markov Processes, Vols. 1-2. Springer-Verlag, Berlin-Göttingen-Heidelberg (1965).
[12] W. Feller, Two singular diffusion problems. Ann. Math. (2) 54 (1951) 173-182.
[13] C. Granger, Long-memory relationship and aggregation of dynamic models. J. Econometrics 14 (1980) 227-238.
[14] R.C. Griffiths, Infinitely divisible multivariate gamma distributions. Sankhya Ser. A 32 (1970) 393-404.
88
E. IGLÓI AND C. TERDIK
[15] E. Iglói, Long-range dependent processes with real bispectrum are third order nonlinear, Technical Report 259 (2001/1).
University of Debrecen, Institute of Mathematics and Informatics (2001).
[16] E. Iglói and G. Terdik, Long-range dependence through Gamma-mixed Ornstein–Uhlenbeck process. Electron. J. Probab.
(EJP) 4 (1999) 1-33.
[17] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes. North-Holland Publishing Co.,
Amsterdam (1981).
[18] S. Jaffard, The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 (1999) 207-227.
[19] S. Karlin and J. McGregor, Classical diffusion processes and total positivity. J. Math. Anal. Appl. 1 (1960) 163-183.
[20] K. Kawazu and S. Watanabe, Branching processes with immigration and related limit theorems. Teor. Verojatnost. Primenen. 16 (1971) 34-51.
[21] A.N. Kolmogorov, Über die analitischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 (1931) 415-458.
[22] A.N. Kolmogorov, Wiener spiral and some other interesting curve in Hilbert space. C. R. Acad. Sci. URSS 26 (1940)
115-118.
[23] A.N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C.
R. Acad. Sci. URSS 30 (1941) 301-305.
[24] P.R. Krishnaiah and M.M. Rao, Remarks on a multivariate gamma distribution. Amer. Math. Monthly 68 (1961) 342-346.
[25] A.S. Krishnamoorthy and M. Parthasarathy, A multi-variate gamma-type distribution. Ann. Math. Statist. 22 (1951)
549-557.
[26] W.E. Leland, M.S. Taqqu, W. Willinger and D.W. Wilson, On the self-similar nature of Ethernet traffic (extended version).
IEEE/ACM Trans. Networking 2 (1994) 1-15.
[27] S.B. Lowen and M.C. Teich, Fractal renewal processes generate 1/f noise. Phys. Rev. E 47 (1993) 992-1001.
[28] B.B. Mandelbrot, Long-run linearity, locally Gaussian processes, h-spectra and infinite variances. Technometrics 10 (1969)
82-113.
[29] B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10
(1968) 422-437.
[30] A.P. Prudnikov, Y.A. Britshkov and O.I. Maritshev, Integrali i ryadi. Nauka, Moscow. Appl. Math. (1981), in Russian.
[31] H.R. Riedi, S.M. Crouse, J.V. Ribeiro and G.R. Baraniuk, A multifractal wavelet model with application to network traffic.
IEEE Trans. Inform. Theory 45 (1999) 992-1018.
[32] B. Ryu and S.B. Lowen, Point process models for self-similar network traffic, with applications. Comm. Statist. Stochastic
Models 14 (1998) 735-761.
[33] G. Samorodnitsky and M.S. Taqqu, Linear models with long-range dependence and finite or infinite variance, edited by
D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt and M.S. Taqqu, New Directions in Time Series Analysis,
Part II. Springer-Verlag, New York, IMA Vol. Math. Appl. 46 (1992) 325-340.
[34] T. Shiga and S. Watanabe, Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete
27 (1973) 37-46.
[35] A.N. Shiryaev, Probability. Springer-Verlag, New York (1996).
[36] S.E. Shreve, Steven shreve’s lectures on stochastic calculus and finance, Online: www.cs.cmu.edu/chal/shreve.html
[37] Y.G. Sinai, Self-similar probability distributions. Theor. Probab. Appl. 21 (1976) 64-84.
[38] G. Szegő, Orthogonal polynomials, in Colloquium Publ., Vol. XXIII. American Math. Soc., New York (1936).
[39] M.S. Taqqu, V. Teverovsky and W. Willinger, Is network traffic self-similar or multifractal? Fractals 5 (1997) 63-73.
[40] G. Terdik, Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis; A Frequency Domain
Approach. Springer-Verlag, New York, Lecture Notes in Statist. 142 (1999).
[41] G. Terdik, Z. Gál, S. Molnár and E. Iglói, Bispectral analysis of traffic in high speed networks. Comput. Math. Appl. 43
(2002) 1575-1583.
[42] W. Willinger, M.S. Taqqu, R. Sherman and D.V. Wilson, Self-similarity through high variability: Statistical analysis of
Ethernet LAN traffic at the source level. Comput. Commun. Rev. 25 (1995) 100-113.
[43] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11
(1971) 155-167.